Question

examples: answer each of the following regarding special angle pairs. 4) use the figure to answer the following. a) how many pairs of adjacent angles are there? list two pairs. b) identify two pairs of vertical angles. c) how many linear pairs are there? list each pair. d) find m∠ebc. e) find m∠abe. 5) a city planner is designing a park. he wants to place two pathways that intersect near the center of the park. if m∠ged = 88°, identify the true statement(s). a) m∠def = 92° b) m∠deg = 92° c) m∠feh = 88° d) m∠deh = 92° e) m∠geh = 88° 6) find x. 7) find x. 9) find x.

Explanation:

Step1: Recall adjacent - angles definition

Adjacent angles share a common side and a common vertex. In the given figure with the intersection of lines, there are 4 pairs of adjacent angles. Two pairs are $\angle ABE$ and $\angle EBC$, $\angle EBC$ and $\angle CBD$.

Step2: Recall vertical - angles definition

Vertical angles are opposite each other when two lines intersect. Two pairs of vertical angles are $\angle ABE$ and $\angle CBD$, $\angle EBC$ and $\angle ABD$.

Step3: Recall linear - pairs definition

A linear pair is a pair of adjacent angles whose non - common sides are opposite rays. There are 2 linear pairs: $\angle ABE$ and $\angle EBC$, $\angle ABD$ and $\angle DBC$.

Step4: Find $m\angle EBC$

Since $\angle ABC$ is a straight - line angle ($180^{\circ}$) and $\angle ABE = 138^{\circ}$, then $m\angle EBC=180^{\circ}-138^{\circ}=42^{\circ}$ (because $\angle ABE+\angle EBC = 180^{\circ}$).

Step5: Find $m\angle ABE$

Given that $\angle ABE = 138^{\circ}$.

For the park - pathway problem:
If $\angle GED = 88^{\circ}$, vertical angles are equal. $\angle FEH$ and $\angle GED$ are vertical angles, so $m\angle FEH = 88^{\circ}$. Also, $\angle DEG$ and $\angle GED$ form a straight - line angle. If $\angle GED = 88^{\circ}$, then $\angle DEG=180^{\circ}-88^{\circ}=92^{\circ}$.

For the angle - finding problems with $x$:

Problem 6

If the sum of the two non - overlapping angles $(2x + 1)^{\circ}$ and $31^{\circ}$ is a certain angle relationship (assuming they are adjacent and form a larger angle), and if they are part of a linear pair or some other defined relationship. If they are adjacent and the sum is $90^{\circ}$ (complementary), then $(2x + 1)+31 = 90$.
$2x+32 = 90$.
$2x=90 - 32=58$.
$x = 29$.

Problem 7

If the two angles $(2x + 10)^{\circ}$ and $110^{\circ}$ are vertical angles, then $2x+10 = 110$.
$2x=110 - 10 = 100$.
$x = 50$.

Problem 9

If the two angles $(x + 12)^{\circ}$ and $90^{\circ}$ are part of a linear pair, then $x + 12+90=180$.
$x+102 = 180$.
$x=180 - 102 = 78$.

Answer:

a) 4 pairs; $\angle ABE$ and $\angle EBC$, $\angle EBC$ and $\angle CBD$
b) $\angle ABE$ and $\angle CBD$, $\angle EBC$ and $\angle ABD$
c) 2 pairs; $\angle ABE$ and $\angle EBC$, $\angle ABD$ and $\angle DBC$
d) $42^{\circ}$
e) $138^{\circ}$

1. b) $m\angle DEG = 92^{\circ}$, c) $m\angle FEH = 88^{\circ}$
2. $x = 29$
3. $x = 50$
4. $x = 78$