Angle relationships: vertical angles, parallel lines, triangle sum

About this lesson

Integrated Math I › Transformations and congruence

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Angle relationships: vertical angles, parallel lines, triangle sum

Warm-up

Three quick questions from earlier lessons.

Discover

Two lines cross, making four angles. One is 70∘70^\circ. What is the angle opposite it?

  1. 70∘70^\circ
  2. 110∘110^\circ

Drag the corners of the triangle. Watch the three angles.

  1. They always add up to 180°
  2. It depends on the shape of the triangle

Key idea

Angle facts, proved with rigid transformations.

Vertical angles are equal. Angles on a straight line add to 180∘180^\circ. With a transversal across parallel lines: corresponding and alternate interior angles are equal; same-side interior angles add to 180∘180^\circ. The angles of a triangle add to 180∘180^\circ.

Worked example

A triangle has angles xx, 2x2x and 60∘60^\circ.

x+2x+60=180x + 2x + 60 = 180
Angle sum.
3x=1203x = 120, x=40x = 40
Solve.
40∘40^\circ, 80∘80^\circ, 60∘60^\circ
The angles.

Your turn, guided

Your turn: a straight angle is split into 115∘115^\circ and yy.

y=65∘y = 65^\circ
180 − 115.

Practice ★

  1. Two angles of a triangle measure 32∘32^\circ and 20∘20^\circ. Find the third angle, in degrees.
  2. Two angles of a triangle measure 50∘50^\circ and 35∘35^\circ. Find the third angle, in degrees.
  1. Two lines cross. One angle measures 30∘30^\circ. Find the angle vertically opposite it, in degrees.

Watch out!

Practice ★★

  1. A transversal crosses two parallel lines and makes a 33∘33^\circ angle with the first. Find the co-interior (same-side interior) angle it makes with the second line, in degrees.
  2. A transversal crosses two parallel lines and makes a 80∘80^\circ angle with the first. Find the co-interior (same-side interior) angle it makes with the second line, in degrees.

Exit ticket

Three questions to check you have got it.

  1. Two angles of a triangle measure 32∘32^\circ and 20∘20^\circ. Find the third angle, in degrees.
  2. Two angles of a triangle measure 50∘50^\circ and 35∘35^\circ. Find the third angle, in degrees.
  3. Two angles of a triangle measure 41∘41^\circ and 80∘80^\circ. Find the third angle, in degrees.

Summary card

Angle relationships

Vertical: equal. Straight line: 180°. Parallel lines: corresponding/alternate equal, same-side interior 180°. Triangle: 180°.

Watch out: equal or supplementary? Picture the rotation.

End of the lesson

Next lesson: Congruence and corresponding parts Back to the chapter

Answers: Angle relationships: vertical angles, parallel lines, triangle sum

A

  1. 128128
  2. 9595

B

  1. 3030

C

  1. 147147
  2. 100100

D

  1. 128128
  2. 9595
  3. 5959

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