Question

in 6 - 8, determine whether each conjecture is true or false. give a counterexample for any false conjecture. 6. given: ∠abc and ∠cbd form a linear pair. conjecture: ∠abc≅∠cbd 7. given: ∠1 and ∠2 are adjacent angles. conjecture: ∠1 and ∠2 form a linear pair. 8. given: (overline{gh}) and (overline{jk}) form a right angle and intersect at (p). conjecture: (overline{gh}perpoverline{jk})

Explanation:

Step1: Recall linear - pair property

A linear pair of angles sum to 180°. Just because ∠ABC and ∠CBD form a linear pair does not mean they are congruent. For example, ∠ABC = 30° and ∠CBD = 150°.

Step2: Recall adjacent - angles and linear - pair relationship

Adjacent angles share a common side and a common vertex. But not all adjacent angles form a linear pair. For example, two adjacent angles in a polygon may not be supplementary.

Step3: Recall perpendicular - lines definition

If two lines (or line - segments) form a right angle, by definition, they are perpendicular.

Answer:

1. False. Counter - example: ∠ABC = 30°, ∠CBD = 150°
2. False. Counter - example: Two adjacent angles in a polygon that are not supplementary.
3. True.