A Tale of Three Intersecting Lines · Lesson 7 of 10
Angle Sum Property and Exterior Angles
“Use a parallel line to explain the 180° angle sum and the exterior-angle relationship.”
• Find a third angle from two known interior angles. • Prove the triangle angle sum using a parallel line and alternate angles. • Distinguish a paper-folding check from a general geometric proof. • Identify an exterior angle and its adjacent interior angle. • Explain and use the exterior angle as the sum of the two non-adjacent interior angles.
Finding the Third Angle
Constructing triangles with the same two angles and different base lengths suggests something interesting: the third angle stays the same. Measurements may differ slightly because a pencil line has thickness or a protractor is hard to align. To know the exact relationship for every triangle, we need reasoning that does not depend on measurement.
Take ΔABC with ∠B = 50° and ∠C = 70°. Draw a straight line XY through A parallel to BC. The sides AB and AC now each cross two parallel lines. This creates angles at A that match the two known base angles.
AB is a transversal, so ∠XAB equals ∠ABC = 50° as alternate angles. AC is another transversal, so ∠CAY equals ∠BCA = 70°. Along the lower side of the straight line XY, the three angles ∠XAB, ∠BAC, and ∠CAY together make a straight angle of 180°.
Problem
Find ∠A when ∠B = 50° and ∠C = 70°.
- 1.Draw XY through A parallel to BC. Alternate angles give ∠XAB = 50° and ∠CAY = 70°.
- 2.The angles along the straight line give 50° + ∠A + 70° = 180°.
- 3.Combine the known angles: 120° + ∠A = 180°.
- 4.Subtract 120° from 180° to obtain ∠A = 60°.
Angle Sum Property
Nothing in the parallel-line argument required the particular values 50° and 70°. In any triangle, the two base angles equal the corresponding alternate angles beside the top angle. Those three angles make a straight angle. This establishes a relationship for every triangle, whatever its side lengths or orientation.
The three interior angles of every triangle add to 180°.
To find one unknown angle, first add the two known angles. Subtract that total from 180°. The result must be positive for a genuine triangle. If the known angles already total 180° or more, they cannot be interior angles of a triangle.
Problem
Find the third angle when two angles are 150° and 15°.
- 1.Add the known angles: 150° + 15° = 165°.
- 2.Subtract their sum from 180°: 180° − 165° = 15°.
- 3.The three angles are 150°, 15°, and 15°; their total is 180°. The very large angle does not prevent a triangle when the other angles are small enough.
Problem
In ΔABC, ∠A = 50° and ∠B = ∠C. Find ∠B and ∠C.
- 1.The two unknown angles together total 180° − 50° = 130°.
- 2.They are given to be equal, so each is half of 130°.
- 3.∠B = ∠C = 65°. Check: 50° + 65° + 65° = 180°.
If all three angles are equal, they share 180° equally and each is 60°. Three angles of 70° would total 210° and are impossible. If only two angles are 70°, the third is 40°. Be precise about which angles are equal rather than assuming a visually balanced picture tells you the answer.
Cut a paper triangle. Fold the top vertex down to the base so the crease is parallel to the base, then fold the two base corners inward so their angles meet the folded top angle. Arrange the three corners together along a straight edge: their angles make a straight angle. This is a useful visual check. The parallel-line argument explains why the result holds for every triangle.
The parallel-line proof is found in Euclid’s Elements, associated with the Greek mathematician who lived around 300 BCE. Adding one well-chosen line can make a relationship visible that was hard to see in the original drawing.
Exterior Angles
Extend one side of a triangle beyond a vertex. The extension and the other side at that vertex form an exterior angle. This outside angle has a useful relationship both with its adjacent interior angle and with the two interior angles at the other vertices.
An angle formed by one side of a triangle and the extension of an adjacent side beyond their common vertex.
In the figure, ∠ACB and ∠ACD add to 180° because CB and CD point in opposite directions along one straight line. The triangle angle sum also gives ∠A + ∠B + ∠ACB = 180°. Both sums equal the same total and contain the same interior angle at C.
Subtract ∠ACB from both relationships. The exterior angle ∠ACD is therefore ∠A + ∠B. These are called the non-adjacent interior angles: they are at the other two vertices, rather than the angle next to the exterior angle.
Problem
In ΔABC, ∠A = 50° and ∠B = 60°. BC is extended through C to D. Find ∠ACD.
- 1.First find the interior angle at C: 180° − (50° + 60°) = 70°.
- 2.The exterior angle and this interior angle form 180°, so ∠ACD = 180° − 70° = 110°.
- 3.Alternatively, add the two non-adjacent interior angles: 50° + 60° = 110°.
- 4.The two methods agree because both follow from the same straight-angle and triangle-angle relationships.
For an exterior angle at C, add the interior angles at A and B. Do not add the exterior angle to all three interior angles and expect a total of 180°. It is the adjacent interior-exterior pair that sums to 180°.
The angle sum and exterior-angle rules are connected, not separate facts to memorise. A parallel line establishes the interior total. Extending a side adds a straight-angle pair, which lets us derive the exterior relationship. Keeping the diagram beside the calculation helps you choose the correct angles.
Quiz
Two interior angles are 36° and 72°. What is the third?
Why do the two base angles appear beside the top angle in the parallel-line proof?
If all three interior angles are equal, what is each angle?
∠A = 50° and ∠B = ∠C. What is ∠B?
An exterior angle at C has non-adjacent interior angles 45° and 65°. What is its measure?
An exterior angle is 125°. What is its adjacent interior angle?
Practice Problems
- Find the third angle for pairs 36°, 72°; 150°, 15°; 90°, 30°; and 75°, 45°. Check each total.
- Draw a triangle and a parallel line through its top vertex. Label the alternate-angle equalities and use them to explain the 180° sum.
- Explain why three 70° angles are impossible, then find the third angle when only two are 70°.
- A triangle has a 50° angle and two equal other angles. Show the steps to find both unknown angles.
- Extend one side of a triangle whose non-adjacent interior angles are 40° and 75°. Find both the exterior angle and its adjacent interior angle.
- An exterior angle at C is 130° and ∠A = 55°. Find ∠B and the interior angle at C.
- Cut and fold a paper triangle to compare its three corners with a straight angle. Explain how the parallel-line proof supports what you observe.
Key Takeaways
• A parallel line through one vertex gives a proof that the interior angles total 180°. • Find an unknown interior angle by subtracting the other two angles’ sum from 180°. • The third angle does not depend on the included side length when the other two angles are fixed. • An exterior angle and its adjacent interior angle add to 180°. • An exterior angle equals the sum of the two non-adjacent interior angles.