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).

Explanation:

Step1: Recall angle - pair definitions

Adjacent angles share a common side and a common vertex. Vertical angles are non - adjacent angles formed by two intersecting lines. Linear pairs are adjacent angles that form a straight line.

Step2: Identify adjacent angle pairs

Pairs like $\angle ABE$ and $\angle EBC$, $\angle EBC$ and $\angle CBD$ are adjacent angles. There are 4 pairs of adjacent angles: ($\angle ABE,\angle EBC$), ($\angle EBC,\angle CBD$), ($\angle CBD,\angle DBA$), ($\angle DBA,\angle ABE$)

Step3: Identify vertical angle pairs

Vertical angle pairs are $\angle ABE$ and $\angle CBD$, $\angle EBC$ and $\angle DBA$.

Step4: Identify linear - pair angle pairs

Linear - pair angle pairs are ($\angle ABE,\angle EBC$), ($\angle EBC,\angle CBD$), ($\angle CBD,\angle DBA$), ($\angle DBA,\angle ABE$). There are 4 linear - pair angle pairs.

Step5: Find angle measures

Since $\angle EBC = 138^{\circ}$, and $\angle ABE$ and $\angle EBC$ form a linear pair, $\angle ABE=180 - 138=42^{\circ}$

Answer:

a) 4 pairs of adjacent angles: ($\angle ABE,\angle EBC$), ($\angle EBC,\angle CBD$), ($\angle CBD,\angle DBA$), ($\angle DBA,\angle ABE$)
b) 2 pairs of vertical angles: ($\angle ABE,\angle CBD$), ($\angle EBC,\angle DBA$)
c) 4 linear - pair angle pairs: ($\angle ABE,\angle EBC$), ($\angle EBC,\angle CBD$), ($\angle CBD,\angle DBA$), ($\angle DBA,\angle ABE$)
d) $m\angle EBC = 138^{\circ}$
e) $m\angle ABE = 42^{\circ}$