angles and triangles homework 4

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4th Grade Geometry Classifying and Measuring Angles

Welcome to the Math Salamanders 4th Grade Geometry worksheets. Here you will find a range of printable fourth grade geometry worksheets, which will help your child to learn to classify and measure angles using a protractor.

There are also some worksheets which explore the properties of a range of 2d shapes.

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4th Grade Geometry

On this webpage you will find our range of free printable geometry worksheets for 4th graders.

There is a range of printable geometry sheets, looking at angles in a variety of forms such as right angles, acute, obtuse and reflex angles.

Using these sheets will help your child to:

• learn how to classify an angle - right, acute, obtuse, reflex;
• learn how to measure angles accurately with a protractor;
• classify different types of triangles - acute, obtuse and right;
• know that angles in a triangle must add up to 180°

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• Try our NEW quick quiz at the bottom of this page.

Classifying Angles

• Angle Classification 1
• Sheet 1 Answers
• PDF version
• Angle Classification 2
• Sheet 2 Answers
• Angle Classification 3
• Sheet 3 Answers

Triangle Classification

• Triangle Classification 1
• Triangle Classification 2

Angle Measuring

• Angle Measuring 1
• Angle Measuring 2
• Angle Measuring 3
• Angle Measuring 4
• Sheet 4 Answers
• Angle Measuring 5
• Sheet 5 Answers

Exploring Deeper Section

The worksheets here involve children applying their knowledge and understanding about 2d shapes.

They are a great way to stretch the more able learners, or get children to think hard about the properties shapes have.

Although the sheets in this section are aimed at 4th grade, they could easily be used by older children.

The first three sheets are in our 3rd Grade Geometry section.

There are UK versions of these sheets due to the different definitions of trapezoid and trapezium!

• Exploring 2d Shapes Sheet 4
• Exploring 2d Shapes Sheet 4 UK version
• UK version Answers
• Exploring 2d Shapes Sheet 5
• Exploring 2d Shapes Sheet 5 UK Version
• UK Version Answers

Looking for some easier Geometry Worksheets

Here is a range of free geometry worksheets for 3rd graders.

• recognize and identify a range of 2d and 3d shapes;
• recognise and identify right angles and lines of symmetry;
• recognise and identify parallel lines;
• identify the faces, edges and vertices of 3d shapes;

The sheets in this section are at a simpler level than those on this page.

• 3rd Grade Geometry Worksheets

Looking for some harder Geometry Worksheets

Here is our selection of 5th grade Geometry worksheets about angles.

• learn how to find a missing angle;
• know that angles inside a right angle must add up to 90°
• know that angles on a straight line must add up to 180°
• know that angles around a point must add up to 360°

The sheets in this section are at a more advanced level than those on this page.

• 5th Grade Geometry Missing Angles

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Area & Perimeter Worksheets 3rd and 4th grade

Here is our range of area and perimeter worksheets.

• know what area and perimeter are;
• know how to find the area and perimeter of rectangles;
• know how to find the area and perimeter of rectilinear shapes;

All the sheets in the section below support elementary math benchmarks.

• Area Worksheets
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Classifying Angles and Triangles Online Quiz

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This quick quiz tests your knowledge of angles and types of triangle.

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Classifying angles

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Geometry worksheets: classifying acute, obtuse and right angles

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Printable Triangle Worksheets

This extensive collection of worksheets on triangles for grades 3 through high-school is incredibly useful in imparting a clear understanding of a variety of topics like classifying triangles, similar triangles, congruence of triangles, median and centroid of a triangle, inequality theorem, Pythagorean inequalities, area, perimeter and angles in a triangle and much more.

List of Triangle Worksheets

Coloring, Tracing, and Drawing Triangles

• Classifying Triangles
• Perimeter of a Triangle
• Area of a Triangle
• Angles in a Triangle
• Medians and Centroid of a Triangle
• Triangle Inequality Theorem
• Pythagorean Inequality Theorem
• Congruent Triangles
• Similar Triangles
• Ordering the Sides and Angles of a Triangle

Explore the Triangle Worksheets in Detail

Take kiddos into the world of triangles with this compilation of worksheets featuring exercises like identifying triangles in real-life, coloring, tracing, and drawing triangles, etc., and a lot more fascinating activities.

Classifying Triangles Worksheets

This cluster of classifying triangles worksheets consists of skills like identifying triangles based on the sides as isosceles, equilateral and scalene and based on angles as acute, obtuse and equilateral. Find exercises to classify triangles offering a mix of angles and sides.

Perimeter of a Triangle Worksheets

Bolster skills with worksheets on finding the perimeter of a triangle with dimensions presented as integers, decimals and fractions. Find the perimeter of scalene, isosceles and equilateral triangles, determine the measure of the sides using the perimeter and solve word problems.

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Angles in a Triangle Worksheets

Featuring myriad exercises, this set of angles in a triangle worksheets help learn the application of angle sum property and exterior angle theorem to find the indicated angles with whole numbers and algebraic expressions. Solve for 'x' and try a set of challenging problems as well.

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Curriculum  /  Math  /  4th Grade  /  Unit 8: Shapes and Angles  /  Lesson 1

Shapes and Angles

Lesson 1 of 18

Criteria for Success

Tips for teachers, anchor tasks.

Problem Set

Target Task

Additional practice.

Lesson Notes

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Identify and draw points, lines, line segments, rays, and angles. 

Common Core Standards

Core standards.

The core standards covered in this lesson

4.G.A.1 — Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Measurement and Data

4.MD.C.5 — Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

Foundational Standards

The foundational standards covered in this lesson

3.G.A.1 — Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Label geometric objects in order to make it easier for them to be identified, located, and differentiated.
• Understand that a point is a precise location in space. It has no dimension.
• Understand that a line is a straight path with no thickness, extending in both directions infinitely. It is one-dimensional. The notation for a line through points $$A$$ and $$B$$ is $$\overleftrightarrow{AB}$$ .
• Understand that a line segment is a part of a line that has two endpoints. It does not extend in either direction infinitely. The notation for a line segment through points $$C$$ and $$D$$ is $$\overline{CD}$$ .
• Understand that a ray is a part of a line that has one endpoint. It extends in one direction infinitely. The notation for a ray through points $$E$$ and  $$F$$ is $$\overset{\rightharpoonup}{EF}$$ or $$\overrightarrow{EF}$$ .
• Understand that an angle is the union of two rays that share a common endpoint. The notation for an angle formed by rays  $$\overrightarrow{HG}$$  and  $$\overrightarrow{HJ}$$ that meet at the point  $$H$$ is $$\angle GHJ$$ .
• Understand that the vertex of an angle is the point where the two rays meet.
• Draw points, lines, line segments, rays, and angles using a ruler or straightedge (MP.5).
• Identify points, lines, line segments, rays, and angles using a ruler or straightedge in two-dimensional figures (MP.2).

Suggestions for teachers to help them teach this lesson

In Criteria for Success #5 you will notice there are two different notations for a ray. Students are welcome to use either notation, but for the remainder of this unit we will use the notation  $$\overrightarrow{AB}$$ .

Lesson Materials

• Blank Paper (1 sheet per student)
• Ruler (1 per student) — This can be any tool used to draw a straight line, e.g., a straightedge, a ruler, etc.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

25-30 minutes

Silently pick a point from the following image:

Write a description of where your point is located.

Guiding Questions

Student response.

Upgrade to Fishtank Plus to view Sample Response.

Pick a Point by Dan Meyer is made available at  dy/dan  under the CC BY 4.0 license . Accessed Feb. 12, 2018, 1:29 p.m..

a.   Draw two points and label them $$A$$ and $$B$$ .

b.   Draw a line between the two points. Label it $$\overleftrightarrow{AB}$$ .

c.   Draw another point that is not on $$\overleftrightarrow{AB}$$ and label it $$C$$ .

d.   Draw a line segment between $$A$$ and $$C$$ . Label it $$\overline{AC}$$ .

e.   Draw another point that is not on $$\overleftrightarrow{AB}$$ or $$\overline{AC}$$ and label it $$D$$ .

f.   Draw a ray from $$B$$ to $$D$$ and label it $$\overrightarrow{BD}$$ .

g.   Draw a point that is not on any of the existing objects. Label it $$E$$ .

h.   Draw an angle between $$D$$ , $$B$$ , and $$E$$ , and label it $$\angle DBE.$$

Grade 4 Mathematics > Module 4 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

Identify any points, lines, line segments, rays, and angles in the following figures.

15-20 minutes

• Problem Set Answer Key

Discussion of Problem Set

• Compare your figure to your partner’s for Problem #1. How are they alike? How are they different? 
• A point indicates a precise location with no size, only position. Points are infinitely small. Why do we mark them with a dot? Won’t our pencil marks have width? Won’t our pencil marks actually cover many points since the dots we draw have width and points do not?
• Just like a point, a line has no thickness. Can we draw a line that has no thickness, or will we always have to imagine that particular attribute? Why do we draw it on paper with thickness?
• How many corners does a triangle have? A square? A quadrilateral? How does that relate to the number of angles each of them has?
• How are a ray and a line similar? How are they different?
• How are angles formed? Where have you seen angles before? 
• Why is it hard to find real-life examples of lines, points, and rays?
• How does your understanding of a number line connect to this lesson on lines?

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

Match each term in Column A with a corresponding picture in Column B. One item is missing in each column, which should be filled in to create the remaining pairs. 

What makes a line different from a line segment? 

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

• Extra Practice Problems Answer Key

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Topic A: Lines and Angles

4.G.A.1 4.MD.C.5

Identify and draw right, obtuse, and acute angles. 

Identify and draw perpendicular lines.

Identify and draw parallel lines.

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Topic B: Measures of Angles

Use iteration with one physical nonstandard unit to measure benchmark angles.

4.MD.C.5 4.MD.C.6

Understand that an angle that turns through  $${{1\over360}}$$ of a circle is called a “one-degree angle” and can be used to measure angles. Use a makeshift protractor to measure benchmark angles. 

Measure angles that are less than 180° using a 180° protractor. Distinguish angle measure from other attributes.

Sketch angles of specified measure using a 180° protractor.

Identify and measure angles as turns and recognize them in various contexts.

Topic C: Measures of Adjacent Angles

Decompose angles using pattern blocks.

Solve problems involving the measure of two adjacent angles.

Solve problems involving the measure of more than two adjacent angles.

Measure and sketch angles that are greater than 180°.

4.MD.C.5 4.MD.C.6 4.MD.C.7

Topic D: Shapes and Lines of Symmetry

Analyze and classify triangles based on side length, angle measure, or both.

4.G.A.1 4.G.A.2

Draw triangles from given criteria.

Analyze and classify quadrilaterals based on parallel and perpendicular lines and the presence or absence of angles of a specified size.

Draw quadrilaterals from given criteria. 

Identify and draw lines of symmetry in two-dimensional figures. 

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• Review of equations
• Simplifying square roots
• Adding and subtracting square roots
• Multiplying square roots
• Dividing square roots
• Line segments and their measures inches
• Line segments and their measures cm
• Segment Addition Postulate
• Angles and their measures
• Classifying angles
• Naming angles
• The Angle Addition Postulate
• Angle pair relationships
• Understanding geometric diagrams and notation
• Parallel lines and transversals
• Proving lines parallel
• Points in the coordinate plane
• The Midpoint Formula
• The Distance Formula
• Parallel lines in the coordinate plane
• Classifying triangles
• Triangle angle sum
• The Exterior Angle Theorem
• Triangles and congruence
• SSS and SAS congruence
• ASA and AAS congruence
• SSS, SAS, ASA, and AAS congruences combined
• Right triangle congruence
• Isosceles and equilateral triangles
• Midsegment of a triangle
• Angle bisectors
• The Triangle Inequality Theorem
• Inequalities in one triangle
• Classifying quadrilaterals
• Angles in quadrilaterals
• Properties of parallelograms
• Properties of trapezoids
• Properties of rhombuses
• Properties of kites
• Areas of triangles and quadrilaterals
• Introduction to polygons
• Polygons and angles
• Areas of regular polygons
• Solving proportions
• Similar polygons
• Using similar polygons
• Similar triangles
• Similar right triangles
• Proportional parts in triangles and parallel lines
• The Pythagorean Theorem and its Converse
• Multi-step Pythagorean Theorem problems
• Special right triangles
• Multi-step special right triangle problems
• Trig. ratios
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• Solving right triangles
• Multi-step trig. problems
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• Permutations
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• Permutations vs combinations
• Probability using permutations and combinations
• Line segments
• Perpendicular segments
• Medians of triangles
• Altitudes of triangles

Chapter 2: Trigonometric Ratios

Exercises: 2.3 Solving Right Triangles

Suggested Homework Problems

Exercises Homework 2.3

Exercise group.

For Problems 1–4, solve the triangle. Round answers to hundredths.

For Problems 5–10,

• Sketch the right triangle described.
• Solve the triangle.

[latex]A = 42{^o}, c = 26[/latex]

[latex]B = 28{^o}, c = 6.8[/latex]

[latex]B = 33{^o}, a = 300[/latex]

[latex]B = 79{^o}, a = 116[/latex]

[latex]A = 12{^o}, a = 4[/latex]

[latex]A = 50{^o}, a = 10[/latex]

For Problems 11–16,

• Without doing the calculations, list the steps you would use to solve the triangle.

[latex]B = 53.7{^o}, b = 8.2[/latex]

[latex]B = 80{^o}, a = 250[/latex]

[latex]A = 25{^o}, b = 40[/latex]

[latex]A = 15{^o}, c = 62[/latex]

[latex]A = 64.5{^o}, c = 24[/latex]

[latex]B = 44{^o}, b = 0.6[/latex]

For Problems 17–22, find the labeled angle. Round your answer to tenths of a degree.

For Problems 23–28, evaluate the expression and sketch a right triangle to illustrate.

[latex]\sin^{-1} 0.2[/latex]

[latex]\cos^{-1} 0.8[/latex]

[latex]\tan^{-1} 1.5[/latex]

[latex]\tan^{-1} 2.5[/latex]

[latex]\cos^{-1} 0.2839[/latex]

[latex]\sin^{-1} 0.4127[/latex]

For Problems 29–32, write two different equations for the statement.

The cosine of [latex]15 {^o}[/latex] is [latex]0.9659\text{.}[/latex]

The sine of [latex]70 {^o}[/latex] is [latex]0.9397\text{.}[/latex]

The angle whose tangent is [latex]3.1445[/latex] is [latex]65 {^o}\text{.}[/latex]

The angle whose cosine is [latex]0.0872[/latex] is [latex]85 {^o}\text{.}[/latex]

Evaluate the expressions and explain what each means. [latex]\begin{equation*} \sin^{-1} (0.6), (\sin 6{^o})^{-1} \end{equation*}[/latex]

Evaluate the expressions and explain what each means. [latex]\begin{equation*} \cos^{-1} (0.36), (\cos 36{^o})^{-1} \end{equation*}[/latex]

For Problems 35–38,

• Sketch a right triangle that illustrates the situation. Label your sketch with the given information.
• Choose the appropriate trig ratio and write an equation, then solve the problem.

The gondola cable for the ski lift at Snowy Peak is [latex]2458[/latex] yards long and climbs [latex]1860[/latex] feet. What angle with the horizontal does the cable make?

The Leaning Tower of Pisa is [latex]55[/latex] meters in length. An object dropped from the top of the tower lands [latex]4.8[/latex] meters from the base of the tower. At what angle from the horizontal does the tower lean?

A mining company locates a vein of minerals at a depth of [latex]32[/latex] meters. However, there is a layer of granite directly above the minerals, so they decide to drill at an angle, starting [latex]10[/latex] meters from their original location. At what angle from the horizontal should they drill?

The birdhouse in Carolyn’s front yard is [latex]12[/latex] feet tall, and its shadow at [latex]4[/latex] pm is [latex]15[/latex] feet [latex]4[/latex] inches long. What is the angle of elevation of the sun at [latex]4[/latex] pm?

For Problems 39–42,

[latex]a = 18, b = 26[/latex]

[latex]a = 35, b = 27[/latex]

[latex]b = 10.6 , c = 19.2[/latex]

[latex]a = 88, c = 132[/latex]

For Problems 43–48,

• Make a sketch that illustrates the situation. Label your sketch with the given information.
• Write an equation and solve the problem.

The Mayan pyramid of El Castillo at Chichén Itzá in Mexico has [latex]91[/latex] steps. Each step is 26 cm high and 30 cm deep.

• What angle does the side of the pyramid make with the horizontal?
• What is the distance up the face of the pyramid, from base to top platform?

An airplane begins its descent when its altitude is 10 kilometers. The angle of descent should be [latex]3{^o}[/latex] from horizontal.

• How far from the airport (measured along the ground) should the airplane begin its descent?
• How far will the airplane travel on its descent to the airport?

A communications satellite is in a low earth orbit (LOE) at an altitude of 400 km. From the satellite, the angle of depression to earth’s horizon is [latex]19.728{^o}\text{.}[/latex] Use this information to calculate the radius of the earth.

The first Ferris wheel was built for the [latex]1893[/latex] Chicago World’s Fair. It had a diameter of [latex]250[/latex] feet, and the boarding platform, at the base of the wheel, was [latex]14[/latex] feet above the ground. If you boarded the wheel and rotated through an angle of [latex]50{^o}\text{,}[/latex] what would be your height above the ground?

To find the distance across a ravine, Delbert takes some measurements from a small airplane. When he is a short distance from the ravine at an altitude of [latex]500[/latex] feet, he finds that the angle of depression to the near side of the ravine is [latex]56{^o}\text{,}[/latex] and the angle of depression to the far side is [latex]32{^o}\text{.}[/latex] What is the width of the ravine? (Hint: First find the horizontal distance from Delbert to the near side of the ravine.)

The window in Francine’s office is [latex]4[/latex] feet wide and [latex]5[/latex] feet tall. The bottom of the window is 3 feet from the floor. When the sun is at an angle of elevation of [latex]64{^o}\text{,}[/latex] what is the area of the sunny spot on the floor?

Which of the following numbers are equal to [latex]\cos 45{^o}\text{?}[/latex]

• [latex]\displaystyle \dfrac{\sqrt{2}}{2}[/latex]
• [latex]\displaystyle \dfrac{1}{\sqrt{2}}[/latex]
• [latex]\displaystyle \dfrac{2}{\sqrt{2}}[/latex]
• [latex]\displaystyle \sqrt{2}[/latex]

Which of the following numbers are equal to [latex]\tan 30{^o}\text{?}[/latex]

• [latex]\displaystyle \sqrt{3}[/latex]
• [latex]\displaystyle \dfrac{1}{\sqrt{3}}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{3}}{3}[/latex]
• [latex]\displaystyle \dfrac{3}{\sqrt{3}}[/latex]

Which of the following numbers are equal to [latex]\tan 60{^o}\text{?}[/latex]

Which of the following numbers are equal to [latex]\sin 60{^o}\text{?}[/latex]

• [latex]\displaystyle \dfrac{3}{\sqrt{2}}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{3}}{2}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{2}}{3}[/latex]
• [latex]\displaystyle \dfrac{2}{\sqrt{3}}[/latex]

For Problems 53–58, choose all values from the list below that are exactly equal to, or decimal approximations for, the given trig ratio. (Try not to use a calculator!)

[latex]\cos 30{^o}[/latex]

[latex]\sin 45{^o}[/latex]

[latex]\tan 30{^o}[/latex]

[latex]\cos 60{^o}[/latex]

[latex]\sin 90{^o}[/latex]

[latex]\cos 0{^o}[/latex]

Fill in the table from memory with exact values. Do you notice any patterns that might help you memorize the values?

Fill in the table from memory with decimal approximations to four places.

For Problems 61 and 62, compare the given value with the trig ratios of the special angles to answer the questions. Try not to use a calculator.

Is the acute angle larger or smaller than [latex]45{^o}\text{?}[/latex]

• [latex]\displaystyle \sin \alpha = 0.7[/latex]
• [latex]\displaystyle \tan \beta = 1.2[/latex]
• [latex]\displaystyle \cos \gamma = 0.65[/latex]

Is the acute angle larger or smaller than [latex]60{^o}\text{?}[/latex]

• [latex]\displaystyle \cos \theta = 0.75[/latex]
• [latex]\displaystyle \tan \phi = 1.5[/latex]
• [latex]\displaystyle \sin \psi = 0.72[/latex]

For Problems 63–72, solve the triangle. Give your answers as exact values.

• Find the perimeter of a regular hexagon if the apothegm is [latex]8[/latex] cm long. (The apothegm is the segment from the center of the hexagon and perpendicular to one of its sides.)
• Find the area of the hexagon.

Triangle [latex]ABC[/latex] is equilateral, and its angle bisectors meet at point [latex]P\text{.}[/latex] The sides of [latex]\triangle ABC[/latex] are 6 inches long. Find the length of [latex]AP\text{.}[/latex]

Find an exact value for the area of each triangle.

Find an exact value for the perimeter of each parallelogram.

• Find the area of the outer square.
• Find the dimensions and the area of the inner square.
• What is the ratio of the area of the outer square to the area of the inner square?
• Find the area of the inner square.
• Find the dimensions and the area of the outer square.

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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