# Lesson 15

Adding the Angles in a Triangle

Let’s explore angles in triangles.

### 15.1: Can You Draw It?

1. Complete the table by drawing a triangle in each cell that has the properties listed for its column and row. If you think you cannot draw a triangle with those properties, write “impossible” in the cell.
2. Share your drawings with a partner. Discuss your thinking. If you disagree, work to reach an agreement.

acute (all angles acute)                                          right (has a right angle)                                    obtuse (has an obtuse angle)

scalene (side lengths all different)

isosceles (at least two side lengthsare equal)

equilateral (three side lengths equal)

### 15.2: Find All Three

Your teacher will give you a card with a picture of a triangle.

1. The measurement of one of the angles is labeled. Mentally estimate the measures of the other two angles.

2. Find two other students with triangles congruent to yours but with a different angle labeled. Confirm that the triangles are congruent, that each card has a different angle labeled, and that the angle measures make sense.
3. Enter the three angle measures for your triangle on the table your teacher has posted.

### 15.3: Tear It Up

Your teacher will give you a page with three sets of angles and a blank space. Cut out each set of three angles. Can you make a triangle from each set that has these same three angles?

1. Draw a quadrilateral. Cut it out, tear off its angles, and line them up. What do you notice?

2. Repeat this for several more quadrilaterals. Do you have a conjecture about the angles?

### Summary

A $$180^\circ$$ angle is called a straight angle because when it is made with two rays, they point in opposite directions and form a straight line.

If we experiment with angles in a triangle, we find that the sum of the measures of the three angles in each triangle is $$180^\circ$$—the same as a straight angle!

Through experimentation we find:

• If we add the three angles of a triangle physically by cutting them off and lining up the vertices and sides, then the three angles form a straight angle.

• If we have a line and two rays that form three angles added to make a straight angle, then there is a triangle with these three angles.

### Glossary Entries

• alternate interior angles

Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

This diagram shows two pairs of alternate interior angles. Angles $$a$$ and $$d$$ are one pair and angles $$b$$ and $$c$$ are another pair.

• straight angle

A straight angle is an angle that forms a straight line. It measures 180 degrees.

• transversal

A transversal is a line that crosses parallel lines.

This diagram shows a transversal line $$k$$ intersecting parallel lines $$m$$ and $$\ell$$.