Angles, triangles, and quadrilaterals

Advanced Math 7 · Unit 8 · euclidean-angles-polygons · Teacher edition

euclidean-angles-polygons

Name ______________________________Date ______________

1.
Two angles are supplementary. One measures $37^\circ$ . What is the other, in degrees?
Answer: ______________
2.
What is the size of each interior angle of a regular nonagon ( $9$ sides), in degrees?
Answer: ______________
3.
In triangle $ABC$ , side $BC$ is extended past $C$ to a point $D$ . The exterior angle $\angle ACD = 112^\circ$ and $\angle BAC = 47^\circ$ . Find $\angle ABC$ , in degrees.
Answer: ______________
4.
Lines $AB$ and $CD$ are parallel. A transversal meets $AB$ at $P$ and $CD$ at $Q$ . Angles $\angle BPQ$ and $\angle DQP$ lie between the parallel lines on the same side of the transversal (co-interior). If $\angle BPQ = 64^\circ$ , what is $\angle DQP$ ?
1. (A)
  $26^\circ$
2. (B)
  $64^\circ$
3. (C)
  $116^\circ$
4. (D)
  $296^\circ$
5.
The interior angles of a pentagon measure $x^\circ$ , $(x + 10)^\circ$ , $(x + 20)^\circ$ , $(x + 30)^\circ$ , and $(x + 40)^\circ$ . What is the measure of the largest angle, in degrees?
1. (A)
  $92$
2. (B)
  $108$
3. (C)
  $118$
4. (D)
  $128$
5. (E)
  $148$

Answer key — Angles, triangles, and quadrilaterals

1.
143euclidean-angles-polygons-01
$180 - 37 = \mathbf{143}$ degrees. ( $53$ is the complement, not the supplement.)
2.
140euclidean-angles-polygons-02
Exterior angle $= \frac{360}{9} = 40^\circ$ , so interior $= 180 - 40 = \mathbf{140}$ degrees. Same by the sum: $\frac{7 \times 180}{9} = \frac{1260}{9} = 140$ . ✓
3.
65euclidean-angles-polygons-03
$\angle ABC = 112 - 47 = \mathbf{65}$ degrees. Check by the long route: $\angle ACB = 180 - 112 = 68$ , and $47 + 65 + 68 = 180$ . ✓
4.
(C)
$116^\circ$
euclidean-angles-polygons-04
Co-interior angles are supplementary: $\angle DQP = 180 - 64 = \mathbf{116}$ degrees. ( $64^\circ$ would be right for alternate or corresponding angles, not co-interior; $26^\circ$ subtracts from $90^\circ$ ; $296^\circ$ subtracts from $360^\circ$ .)
5.
(D)
$128$
euclidean-angles-polygons-05
$5x + 100 = 540 \Rightarrow x = 88$ . Largest $= 88 + 40 = \mathbf{128}$ degrees. ( $92$ uses $360^\circ$ as the interior sum — the exterior-angle trap; $108$ is a regular pentagon's angle; $118$ stops at $x + 30$ .)