Question

in 6 - 8, determine whether each conjecture is true or false. give a counterexample for any false conjecture.

1. given: ∠abc and ∠cbd form a linear pair.

conjecture: ∠abc≅∠cbd

1. given: ∠1 and ∠2 are adjacent angles.

conjecture: ∠1 and ∠2 form a linear pair.

1. given: (overline{gh}) and (overline{jk}) form a right angle and intersect at (p).

conjecture: (overline{gh}perpoverline{jk})

Explanation:

Step1: Recall linear - pair and congruent - angle definitions

A linear pair of angles are adjacent angles that add up to 180 degrees. Congruent angles have equal measures. Just because two angles form a linear pair doesn't mean they are congruent.
For ∠ABC and ∠CBD forming a linear pair, they are not necessarily congruent. For example, if ∠ABC = 30° and ∠CBD=150°, they form a linear pair but are not congruent. So the conjecture "∠ABC≅∠CBD" is false.

Step2: Recall adjacent - and linear - pair definitions

Adjacent angles share a common side and a common vertex. A linear pair is a special case of adjacent angles where the non - common sides form a straight line and the sum of the angles is 180 degrees. Just because ∠1 and ∠2 are adjacent does not mean they form a linear pair. For example, two adjacent angles in a right - angled corner (e.g., 30° and 60°) are adjacent but not a linear pair. So the conjecture "∠1 and ∠2 form a linear pair" is false.

Step3: Recall perpendicular - line definition

If two lines (or line segments) form a right angle when they intersect, they are perpendicular. Given that $\overline{GH}$ and $\overline{JK}$ form a right angle and intersect at $P$, by the definition of perpendicular lines, $\overline{GH}\perp\overline{JK}$. So the conjecture is true.

Answer:

1. False. Counter - example: ∠ABC = 30°, ∠CBD = 150°
2. False. Counter - example: Two adjacent angles in a right - angled corner (e.g., 30° and 60°)
3. True