Triangle Angle Sum

8.G.5Grade 8Geometry20 practice questions

Every triangle - tall, flat, tiny or huge - has angles that add to exactly 180°. Know two angles and you know the third. The harder skill is turning words into the right equation.

Know these first

Parallel Lines and Transversals 8.G.5
Master List skill 7.G.5

This unlocks

Exterior Angles of Triangles 8.G.5
Angle-Angle (AA) Similarity 8.G.5

What you'll be able to do

Explain why the three angles of a triangle add to 180°.
Find a missing angle, including in right and isosceles triangles.
Turn a description like "twice the smallest" into an equation and solve it.
Use the 180° rule inside bigger figures, like four-sided shapes and parallel lines.

The big idea

Draw any triangle. Through the top corner, draw a line parallel to the bottom side.

The two bottom angles now have twins at the top: each is an alternate interior angle with the new line, so they're equal. The three angles at the top sit side by side on a straight line, and a straight line is 180°. So the three angles of any triangle add to 180°.

Harder problems describe the angles instead of giving them: "one angle is twice another," "the base angles are equal." Name the smallest angle x, write every angle in terms of x, and set the total to 180.

Words you need

Interior angle: An angle inside the triangle, at one of its corners.
Right triangle: One 90° angle, so the other two add to 90°.
Isosceles triangle: Two equal sides. The base angles opposite them are equal too.
Equilateral triangle: All sides equal, so all three angles are 60°.

Worked examples

Example 1

A triangle has angles of 48° and 67°. Find the third angle.

Add the two you know: 48° + 67° = 115°.
Subtract from 180°: 180° − 115° = 65°.

Answer: 65°

Example 2

The angles of a triangle are x°, (x + 20)° and 2x°. Find x and each angle.

Write the 180° rule: x + (x + 20) + 2x = 180.
Combine like terms: 4x + 20 = 180, so 4x = 160 and x = 40.
The angles are 40°, 60° and 80°. Check: 40 + 60 + 80 = 180.

Answer: x = 40; the angles are 40°, 60°, 80°

Example 3

An isosceles triangle has base angles of (3x − 8)° and (x + 22)°. Find x and the top angle.

Base angles of an isosceles triangle are equal: 3x − 8 = x + 22.
Subtract x and add 8: 2x = 30, so x = 15.
Each base angle is 15 + 22 = 37°.
Top angle = 180° − 37° − 37° = 106°.

Answer: x = 15; each base angle is 37° and the top angle is 106°

Common mistakes

"The angles add to 360°." That's the rule for four-sided shapes. A triangle's angles add to 180°.

Answering with x. When the question asks for the angles, find x, then put it back into each expression.

Setting the base angles of an isosceles triangle to add to 180°. Base angles are EQUAL to each other. Set them equal first; the 180° rule finds the top angle after.

Trusting how the drawing looks. A corner that looks like 90° isn't one unless it's marked with a small square or you're told.

Practice: 20 questions

14 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

Find the missing angle.
Find the missing angle.
A triangular garden bed has corners of 52° and 61°. What does the third corner measure? Word problem
A triangular kite sail has angles of 35° and 105°. What is its third angle? Word problem
A ladder leans on a wall and makes a 72° angle with the ground. The wall meets the ground at 90°. What angle does the ladder make with the wall? Word problem

Tier 2 Intermediate One more step to think through

The angles are 2x°, 3x° and 4x°. Find x and each angle.
The angles are (x + 15)°, (x + 25)° and (x − 10)°. Find x and each angle.
A slice of pizza is a triangle with a 40° tip. The two corners along the crust are equal. What does each crust corner measure? Word problem
The angles of a sailboat's triangular sail are x°, 3x° and 5x°. Find each angle. Word problem
A roof truss has a 130° angle at its peak, and its two bottom corners are equal. What does each bottom corner measure? Word problem
A square napkin folded corner to corner makes a triangle with one 90° corner and two equal corners. What does each of the equal corners measure? Word problem
In a right triangle, one sharp angle is 4 times the other. Find both sharp angles. Word problem

Tier 3 Advanced Several steps - find the hidden structure

An isosceles triangle has base angles of (3x + 5)° and (x + 45)°. Find x and the top angle.
In a triangle, the largest angle is twice the smallest, and the middle angle is 12° more than the smallest. Find all three angles. Word problem
An A-frame swing set's side is an isosceles triangle. Its base angles are (2x + 10)° and (4x − 20)°. Find the angle at the top. Word problem
The angles of a triangular park are in the ratio 2 : 3 : 7. Find each angle. Word problem
A four-sided garden bed has angles of 85°, 95° and 110°. A path along one diagonal splits it into two triangles. What is the fourth angle? Word problem

Tier 4 Challenge Stretch problems, some above grade level

In triangle ABC, a line through C is parallel to AB. It makes a 48° angle with side CA and a 63° angle with side CB. Find angle ACB.
In a triangular plot, the largest angle is 20° more than the other two angles added together. What is the largest angle? Word problem
A warning sign is an isosceles triangle with base angles of (x/2 + 30)° and (2x − 15)°. Find the angle at its top. Word problem

Answers

Tier 1 Foundations

55°. The angles of a triangle add to 180°: 180° − 55° − 70° = 55°.
52°. The angles of a triangle add to 180°: 180° − 90° − 38° = 52°.
67°. The angles of a triangle add to 180°: 180° − 52° − 61° = 67°.
40°. The angles of a triangle add to 180°: 180° − 35° − 105° = 40°.
18°. The angles of a triangle add to 180°: 180° − 72° − 90° = 18°.

Tier 2 Intermediate

x = 20; the angles are 40°, 60°, 80°. The three angles add to 180°: (2x) + (3x) + (4x) = 180, so x = 20.
x = 50; the angles are 65°, 75°, 40°. The three angles add to 180°: (x + 15) + (x + 25) + (x - 10) = 180, so x = 50.
x = 70; the angles are 70°, 70°, 40°. The three angles add to 180°: (x) + (x) + (40) = 180, so x = 70.
x = 20; the angles are 20°, 60°, 100°. The three angles add to 180°: (x) + (3x) + (5x) = 180, so x = 20.
x = 25; the angles are 25°, 25°, 130°. The three angles add to 180°: (x) + (x) + (130) = 180, so x = 25.
x = 45; the angles are 45°, 45°, 90°. The three angles add to 180°: (x) + (x) + (90) = 180, so x = 45.
x = 18; the angles are 18°, 72°, 90°. The three angles add to 180°: (x) + (4x) + (90) = 180, so x = 18.

Tier 3 Advanced

x = 20; each base angle is 65° and the top angle is 50°. Base angles of an isosceles triangle are equal: 3x + 5 = x + 45, so x = 20. Top angle = 180° − 2 × 65° = 50°.
x = 42; the angles are 42°, 54°, 84°. The three angles add to 180°: (x) + (x + 12) + (2x) = 180, so x = 42.
x = 15; each base angle is 40° and the top angle is 100°. Base angles of an isosceles triangle are equal: 2x + 10 = 4x − 20, so x = 15. Top angle = 180° − 2 × 40° = 100°.
x = 15; the angles are 30°, 45°, 105°. The three angles add to 180°: (2x) + (3x) + (7x) = 180, so x = 15.
70. The diagonal makes two triangles, so the four angles add to 2 × 180° = 360°. The fourth is 360° − 85° − 95° − 110° = 70°.

Tier 4 Challenge

69. The 48°, the angle ACB and the 63° sit side by side on the straight parallel line, so angle ACB = 180° − 48° − 63° = 69°.
100. Call the other two together s. Then the largest is s + 20, and s + (s + 20) = 180, so s = 80 and the largest is 100°.
x = 30; each base angle is 45° and the top angle is 90°. Base angles of an isosceles triangle are equal: x/2 + 30 = 2x − 15, so x = 30. Top angle = 180° − 2 × 45° = 90°.

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