Derive direct proofs involving angles and triangles

Alberta Grade 11 Mathematics

Geometry

Use known angle facts to build a clear chain of reasons

A proof is a set of statements that shows why a conclusion must be true. In a direct proof, begin with the information given and use accepted geometry facts to reach the conclusion. Each step needs a reason. This lesson reviews the needed angle facts, then uses them to prove results about triangles and angles.

What you will learn

Recognize angle relationships that can be used in a proof.
Write a direct proof by linking a starting fact to a conclusion.
Derive the triangle angle sum and use it to prove other angle facts.

1. Review: angle facts and proof language

An angle measures the turn between two rays that share an endpoint. The endpoint is called the vertex. A straight angle measures

180^\circ

. When two adjacent angles form a straight line, they are supplementary: their measures add to

180^\circ

.

When two lines cross, opposite angles are called vertical angles. Vertical angles have equal measures. When a line crosses a pair of parallel lines, alternate interior angles are equal. These facts let us connect angle measures in a diagram.

A triangle has three sides and three interior angles. Interior angles are the angles inside the triangle. A useful goal in a proof is to show how the given facts force a relationship among those angles.

In a proof, a statement tells what is known or what follows. A reason names the definition or geometry fact that justifies it. For example, if two angles form a straight line, their measures sum to

180^\circ

because angles on a straight line are supplementary.

m\angle A + m\angle B = 180°

A direct proof moves from known information to the desired conclusion.
Give a reason for every new statement.
Use only angle facts that apply to the stated diagram.

2. Build a proof from a diagram

A diagram helps you see possible relationships, but it does not prove them. A sketch may not be drawn to scale. Use the written facts and valid geometry rules, not how large an angle appears.

A helpful proof plan is to mark the given information, identify the conclusion, and look for a rule that links them. Then write one statement at a time. If a step uses an earlier result, state that result as its reason.

For a triangle, a line through one vertex and parallel to the opposite side creates useful equal angles. The equalities come from alternate interior angles. The three angles along the new straight line add to

180^\circ

. Together, these facts prove that the interior angles of a triangle also add to

180^\circ

.

Once a result has been proved, it can support another proof. For example, the triangle angle sum and the straight-angle rule can show that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. An exterior angle is formed by extending one side of a triangle.

m\angle A + m\angle B + m\angle C = 180°

A diagram suggests a plan but is not evidence by itself.
Parallel lines can connect triangle angles to a straight angle.
A proved result may be used in a later proof.

3. Write and check a direct proof

A proof should be clear enough that another student can follow it without guessing. Write the claim, list the facts that apply, and connect each statement to a reason. Avoid jumping from the first line to the conclusion.

Check that the conclusion matches the claim, that angle names refer to the correct parts of the diagram, and that every equation follows from a stated fact. A numerical answer can help check a calculation, but a proof must still explain why the relationship is always true.

For instance, a triangle with two angles measuring

52^\circ

and

71^\circ

has a third angle of

57^\circ

, since the three angles total

180^\circ

. This calculation illustrates the triangle sum; the proof of the triangle sum comes from the parallel-line argument, not from measuring one example.

m\angle C = 180° - m\angle A - m\angle B

Use reasons that match the specific angle relationship.
Separate a general proof from a check using one numerical example.
Finish by stating the conclusion in the terms of the claim.

4. Student try

Try this before reading the examples again: in a triangle, one exterior angle is adjacent to an interior angle measuring

68^\circ

. The two non-adjacent interior angles are

47^\circ

and

x^\circ

. Find

x

and explain why the exterior angle equals the sum of those two non-adjacent angles.

Use the triangle angle sum and the fact that adjacent angles on a straight line sum to

180^\circ

. Keep the two relationships separate before comparing them.

x = 65°

Identify the interior angle adjacent to the exterior angle.
Use the triangle angle sum for the three interior angles.
Use a straight angle for the adjacent exterior and interior angles.

Worked example

Prove the triangle angle sum

In triangle

ABC

, draw a line through

C

parallel to

AB

. Prove that the three interior angles of the triangle sum to

180^\circ

.

Triangle angle-sum construction

A parallel line through the triangle vertex links the interior angles to a straight angle.

Mark the equal angles
Because the new line is parallel to $AB$ , the angle it makes with $AC$ equals interior angle $A$ . These are alternate interior angles. The angle it makes with $BC$ equals interior angle $B$ for the same reason.
$m\angle A = m\angle 1, m\angle B = m\angle 2$
Use the straight angle
The two marked angles and interior angle $C$ lie along a straight line at $C$ . Their measures therefore add to $180^\circ$ .
$m\angle 1 + m\angle C + m\angle 2 = 180°$
Substitute equal measures
Replace the two marked angles with the equal triangle angles from the first step. This proves the required relationship for any triangle, not just a particular drawing.
$m\angle A + m\angle B + m\angle C = 180°$

Answer: The interior angles of triangle

ABC

sum to

180^\circ

.

Check: The proof uses the parallel-line angle relationship and the straight-angle fact. It does not depend on measuring the sketch.

Worked example

Prove the exterior angle relationship

A side of triangle

PQR

is extended past

R

, forming an exterior angle. Prove that the exterior angle equals the sum of the two interior angles at

P

and

Q

.

Exterior angle relationship

The exterior angle and the adjacent interior angle form a straight angle.

Use the triangle sum
The three interior angles of a triangle sum to a straight angle. This follows from the result proved in Example 1.
$m\angle P + m\angle Q + m\angle R = 180°$
Use the adjacent straight angle
The exterior angle and interior angle $R$ share a side and form a straight line. Their measures also sum to $180^\circ$ .
$m\angle R + m\angle E = 180°$
Compare the equations
Both expressions equal $180^\circ$ and include $m\angle R$ . Removing that common angle measure leaves the exterior angle equal to the other two interior angles combined.
$m\angle E = m\angle P + m\angle Q$

Answer: The exterior angle equals the sum of the two non-adjacent interior angles.

Check: The exterior angle is larger than either of the two individual interior angles when both are positive, which is consistent with their sum.

Worked example

Find and justify an unknown angle

In triangle

XYZ

, the angles at

X

and

Y

measure

46^\circ

and

79^\circ

. Side

YZ

is extended past

Z

. Find the exterior angle at

Z

, and justify the result using direct angle relationships.

Interior and exterior angles

The outside angle is supplementary to the triangle's interior angle at Z.

Find the interior angle at Z
The interior angles of a triangle sum to $180^\circ$ . Subtract the two known angles to find the third.
$m\angle Z = 180° - 46° - 79° = 55°$
Find the exterior angle
The exterior angle and the interior angle at $Z$ form a straight angle, so their measures add to $180^\circ$ .
$m\angle E = 180° - 55° = 125°$
Connect to the other interior angles
The result also agrees with the exterior angle relationship proved in Example 2: add the two non-adjacent interior angles.
$46° + 79° = 125°$

Answer: The interior angle at

Z

is

55^\circ

, and the exterior angle is

125^\circ

.

Check: The interior and exterior angles at Z sum to

180^\circ

. The exterior angle also equals the sum of

46^\circ

and

79^\circ

.

Common mistakes and how to avoid them

Saying two angles are equal because they look equal in the drawing.

Correction: Name a fact that proves the equality, such as alternate interior angles formed by parallel lines.

Giving a correct final equation but no reasons for the steps.

Correction: State the applicable rule at each step so the chain of reasoning is complete.

Confusing an exterior angle with the interior angle beside it.

Correction: The adjacent interior and exterior angles form a straight angle and sum to

180^\circ

.

Lesson summary

A direct proof starts with known facts and reaches a conclusion through justified steps.
Angles on a straight line sum to $180^\circ$ ; alternate interior angles formed by parallel lines are equal.
The interior angles of a triangle sum to $180^\circ$ .
A triangle's exterior angle equals the sum of its two non-adjacent interior angles.

Check your understanding

Question 1

A triangle has interior angles of

38^\circ

and

91^\circ

. What is the third interior angle?

$51^\circ$
$129^\circ$
$89^\circ$
$251^\circ$

Show answer and explanation

51^\circ

Subtract the known angles from

180^\circ

:

180^\circ - 38^\circ - 91^\circ = 51^\circ

.

Question 2

A triangle's two non-adjacent interior angles to an exterior angle are

42^\circ

and

67^\circ

. What is the exterior angle?

$109^\circ$
$71^\circ$
$113^\circ$
$247^\circ$

Show answer and explanation

109^\circ

The exterior angle equals the sum of the two non-adjacent interior angles:

42^\circ + 67^\circ = 109^\circ

.

Question 3

Which fact justifies that two alternate interior angles are equal?

The lines crossed by the transversal are parallel.
The angles form a straight line.
The angles are inside a triangle.
The diagram appears symmetrical.

Show answer and explanation

The lines crossed by the transversal are parallel.

When a transversal crosses parallel lines, alternate interior angles are equal.

Key terms

Direct proof: A chain of justified statements that begins with known information and reaches the claim.
Supplementary angles: Two angles whose measures add to $180^\circ$ .
Vertical angles: Opposite angles formed when two lines cross; they have equal measures.
Alternate interior angles: Angles between two lines on opposite sides of a crossing line; they are equal when the two lines are parallel.
Exterior angle: An angle outside a triangle, formed by extending one of its sides.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation G1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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