Question

Use the figure below to find the value of each indicated angle.

3
< 1=
<3=
<5 =
<7 =
<9=
<11 =
4
8
10
7
9
5
59°
6
<2=
<4=
<6 =
<8=
<10 =

Answer 1

• ∠1 = ∠3 = ∠6 = 59°
• ∠2 = ∠4 = ∠5 = ∠11 = 31°
• ∠7 = ∠10 = 118°
• ∠8 = ∠9 = 62°

Explanation:

You want the measures of the angles created by the two diagonals in a rectangle, if the larger of one of the corner angles is 59°.

Angles

Congruent angles are ...

• {1, 3, 6, 59°}
• {2, 4, 5, 11, (90-59)°}
• {8, 9, (180-2·59)°}
• {7, 10, (2·59)°}

Isosceles triangles

All of the triangles formed are isosceles triangles. This means the base angles of each triangle are congruent. Opposite triangles are congruent, so we immediately have sets of 4 angles that are all congruent. These are reflected in the first two sets of congruent angles listed above.

Central angles

The angles at the center of the figure can be found a couple of different ways. Each one is an exterior angle to the two congruent triangles that don't include it. That means its measure is equal to the sum of the base angles of those triangles. The base angles are equal, so we know, for example, that angles 7 and 10 are twice the measure of the 59° angle marked. They are 118°.

Of course, angles 8 and 9 are supplementary to that, or 62°, which is also twice the measure of the base angles 2, 4, 5, and 11, each of which is 31°.

Summary

• ∠1 = ∠3 = ∠6 = 59°
• ∠2 = ∠4 = ∠5 = ∠11 = 31°
• ∠7 = ∠10 = 118°
• ∠8 = ∠9 = 62°

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