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Magazine Chapter 2: Q16. (page 63)
a. Are there any angles in the diagram that must be congruent to∠4? Explain.
b. If ∠4 and ∠5 are supplementary, name all angles shown that must be congruent to ∠4.
Short Answer
a. Yes, there is an angle∠2 that is congruent to∠4 becausethe angles∠2and∠4 are the congruent angles.
b. If∠4and∠5are supplementary, all angles that must be congruent to∠4 are∠8and∠6.
Step by step solution
Part a. Step 1. Observe the given diagram.
The given diagram is:
Prat a. Step 2. Description of step.
From the given diagram it can be noticed that angles ∠2 and ∠4 are the vertically opposite angles.
Therefore, .
That implies that angles ∠2 and ∠4 are congruent angles.
Part a. Step 3. Description of step.
Yes, there is an angle∠2that is congruent to∠4 becausethe angles ∠2 and ∠4 are the congruent angles.
Part b. Step 1. Observe the given diagram.
The given diagram is:
Part b. Step 2. Description of step.
It is being given that ∠4and∠5are supplementary.
Therefore, the sum of angles ∠4and∠5is 180°.
From the given figure it can be noticed that angles ∠4and∠5are adjacent angles on the same side of the transversal.
If two lines are parallel and there is a transversal then the sum of the adjacent angles on the same side of the transversal is 180°.
As angles ∠4and∠5are adjacent angles on the same side of the transversal and the sum of the angles ∠4and∠5is 180°.
Therefore, the two given lines are parallel.
From the given diagram it can be noticed that angles ∠4and∠8are the corresponding angles.
Therefore, .
As the two given lines are parallel and the angles ∠4and∠6are thealternate interior angles.
Therefore, .
Step 3. Description of step.
If∠4and∠5are supplementary, all angles that must be congruent to∠4 are∠8and∠6.
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, then 
is a right angle.