Triangles Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Triangles.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A polygon with exactly three sides and three interior angles that always sum to exactly 180 degrees.

The simplest polygon—you need at least 3 sides to enclose space.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A triangle is a shape whose constraints come from exactly three sides and three angles.

Common stuck point: The procedure for triangles is the easy part; the trap is classifying by sides when the question asks about angles. Asking "Is it a closed polygon with exactly three straight sides?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is it a closed polygon with exactly three straight sides?

Worked Examples

Example 1

easy

Two angles of a triangle measure

50°

and

65°

. Find the third angle.

Find the third angle x.

Answer

x = 65 °

First step

The angle sum property states that all angles in a triangle add up to

180°

Full solution

2
Let the third angle be $x$ : $50 + 65 + x = 180$ .
3
Solve: $x = 180 - 115 = 65°$ .

The triangle angle sum property (

180°

) is one of the foundational facts in geometry. Since two angles are

50°

and

65°

, and the third is also

65°

, this is an isosceles triangle.

Example 2

medium

Classify the triangle with sides

5

cm,

5

cm, and

8

cm by its sides and determine whether it is acute, right, or obtuse.

Example 3

easy

A triangle has angles of

90^\circ

45^\circ

, and

x

. Find

x

and classify the triangle.

Find x and classify the triangle.

Example 4

medium

The angles of a triangle are

(2x)^\circ

(3x)^\circ

, and

(4x)^\circ

. Find

x

and each angle.

With x = 20, angles are 40°, 60°, and 80°.

Example 5

medium

Two sides of a triangle measure

7

and

11

. The third side is an integer. Find all possible values of the third side.

Example 6

medium

Classify the triangle with sides

6, 8, 10

by sides and by angles.

Example 7

medium

In an isosceles triangle, one of the equal angles is

25^\circ

less than the third (vertex) angle. Find all three angles.

Example 8

hard

Triangle

ABC

has

\angle A = 40^\circ

and the bisector of

\angle B

meets

AC

D

so that

\angle BDA = 100^\circ

. Find

\angle C

Example 9

hard

In isosceles triangle

ABC

with

AB = AC

, the vertex angle

\angle A = 36^\circ

. The bisector of

\angle B

meets

AC

D

. Find

\angle BDA

Example 10

hard

In a triangle, one angle is

20^\circ

more than the smallest and the largest is twice the smallest. Find all three angles.

Smallest 40°, middle 60°, largest 80° (twice the smallest).

Example 11

challenge

In triangle

ABC

\angle B = 50^\circ

and

\angle C = 30^\circ

. Point

D

lies on side

BC

such that

AD = BD

. Find

\angle DAC

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

An equilateral triangle has a perimeter of

36

cm. Find the length of each side.

Example 2

medium

The three angles of a triangle are in the ratio

2:3:4

. Find the measure of each angle.

Angles in ratio 2:3:4 are 40°, 60°, and 80°.

Example 3

easy

Two angles of a triangle are

50^\circ

and

60^\circ

. Find the third angle.

Find the third angle x.

Example 4

easy

A triangle has sides

5

5

, and

8

. What type is it (by sides)?

Classify by side lengths: two sides equal 5, base equals 8.

Example 5

easy

Each angle of a certain triangle is

60^\circ

. What type is it?

Example 6

easy

One angle of a triangle is

90^\circ

. What type is it (by angles)?

Example 7

easy

Can a triangle have two right angles?

Example 8

easy

A right triangle has one acute angle of

35^\circ

. Find the other acute angle.

Find the missing acute angle x in this right triangle.

Example 9

easy

Can sides of length

2

3

, and

10

form a triangle?

Example 10

easy

An isosceles triangle has a base angle of

70^\circ

. Find the other base angle.

Find the other base angle x.

Example 11

medium

The apex angle of an isosceles triangle is

40^\circ

. Find each base angle.

Find each base angle x given the apex angle is 40°.

Example 12

medium

The exterior angle of a triangle at one vertex is

120^\circ

. The two non-adjacent interior angles are equal. Find each.

Use the exterior-angle theorem: x + x

Example 13

medium

A triangle has two sides of length

7

and

4

. Between which two whole numbers must the third side lie?

Example 14

medium

Can a triangle be both right and isosceles? If so, find its angles.

A right isosceles triangle has angles 90°, 45°, 45°.

Example 15

medium

The angles of a triangle are in the ratio

1:2:3

. Find all three angles.

Angles in ratio 1:2:3 give the 30-60-90 triangle.

Example 16

medium

An obtuse triangle has one angle of

110^\circ

. The other two are equal. Find them.

Find each base angle x given the obtuse apex is 110°.

Example 17

medium

Why can a triangle never have two obtuse angles?

Example 18

medium

In a triangle, the largest angle is opposite the longest side. If a triangle has sides

6 < 8 < 11

, which angle is largest?

Example 19

challenge

In triangle

ABC

, angle

A = 80^\circ

. The bisectors of angles

B

and

C

meet at point

I

. Find angle

BIC

Example 20

challenge

How many non-congruent triangles have integer side lengths and a perimeter of

12

Example 21

challenge

In triangle

ABC

AB = AC

and angle

A = 100^\circ

. Point

D

lies on

BC

with

BD = AB

. Find angle

DAC

Example 22

challenge

Prove that the three medians of a triangle always divide it into 6 smaller triangles of equal area.

Example 23

easy

A triangle has angles

30^\circ

and

80^\circ

. What is the third angle?

Find the third angle x.

Example 24

easy

Can a triangle have sides of length

4

5

, and

6

Example 25

easy

An isosceles triangle has a vertex angle of

40^\circ

. Find each base angle.

Find each base angle x in this isosceles triangle.

Example 26

easy

True or false: a triangle can have one obtuse angle and one right angle.

Example 27

easy

An equilateral triangle has side length

9

cm. Find its perimeter.

Example 28

medium

In a right triangle, one acute angle is twice the other. Find both acute angles.

One acute angle (60°) is twice the other (30°).

Example 29

medium

An isosceles triangle has a perimeter of

32

cm, and its base is

10

cm. Find the length of each equal leg.

Example 30

medium

In triangle

ABC

\angle A = 2\angle B

and

\angle C = \angle B + 20^\circ

. Find each angle.

∠A

Example 31

medium

A triangle has angles

x

x + 10^\circ

, and

x + 20^\circ

. Is the triangle acute, right, or obtuse?

All angles acute: 50°, 60°, 70°.

Example 32

medium

In a triangle, the longest side is opposite which angle?

Example 33

hard

The sides of a triangle have lengths

x

x+3

, and

x+6

. For what range of

x

is the triangle valid?

Example 34

hard

Two angles of a triangle are

(3x + 10)^\circ

and

(2x - 5)^\circ

, and the third is

(x + 15)^\circ

. Find

x

and the largest angle.

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Related Concepts

Basic Shapes Angles Pythagorean Theorem Triangles Trigonometric Functions

Background Knowledge

These ideas may be useful before you work through the harder examples.

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