Chapter 12.5, Problem 35E

a.

To determine

To show: $△ABD\cong △CBD$ . Also, state which theorem or postulate is used.

Answer to Problem 35E

By HL Congruence Theorem.

Explanation of Solution

Given information:

  

Both spotlights are the same distance from the stage.

Formula used:

The Hypotenuse-Leg Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle , then the two triangles are congruent.

Calculation:

According to the question,

There are two triangles $△ABD\text{ and }△CBD\text{.}$

Now,

Statements	Reasons
H 1. $\overline{AB}=\overline{BC}$ 2. $\overline{BD}\perp \overline{AD}$ ,   $\overline{BD}\perp \overline{DC}$ 3. $\angle ADB\text{ and }\angle CDB\text{ are right angles}\text{.}$ 4. $△ABD\text{ and }△CBD\text{ are right triangles}\text{.}$	1.Given 2. Given 3. Definition of $\perp$ lines. 4. Definition of a right triangle.
L 5. $\overline{BD}=\overline{BD}$ 6. $△ABD\cong △CBD$	5. Reflexive Property of Congruence 6. HL Congruence Theorem

Therefore, the triangles are congruent by HL Congruence Theorem that states - If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle , then the two triangles are congruent.

Hence proved.

b.

To determine

To show: Whether all the four right triangles in the diagram are congruent.

Answer to Problem 35E

Yes, all the four triangles are congruent.

Explanation of Solution

Given information:

  

Both spotlights are the same distance from the stage.

Formula used:

The Hypotenuse-Leg Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle , then the two triangles are congruent.

Calculation:

According to the question,

There are two triangles $△ABD\text{ and }△CBD\text{.}$

Now,

Statements	Reasons
H 1. $\overline{AB}=\overline{BC}$ 2. $\overline{BD}\perp \overline{AD}$ ,   $\overline{BD}\perp \overline{DC}$ 3. $\angle ADB\text{ and }\angle CDB\text{ are right angles}\text{.}$ 4. $△ABD\text{ and }△CBD\text{ are right triangles}\text{.}$	1.Given 2. Given 3. Definition of $\perp$ lines. 4. Definition of a right triangle.
L 5. $\overline{BD}=\overline{BD}$ 6. $△ABD\cong △CBD$	5. Reflexive Property of Congruence 6. HL Congruence Theorem

There are two triangles $△CEG\text{ and }△FEG\text{.}$

Now,

Statements	Reasons
H 1. $\overline{CE}=\overline{EF}$ 2. $\overline{EG}\perp \overline{CG}$ ,   $\overline{EG}\perp \overline{GF}$ 3. $\angle CGE\text{ and }\angle FGE\text{ are right angles}\text{.}$ 4. $△CEG\text{ and }△FEG\text{ are right triangles}\text{.}$	1.Given 2. Given 3. Definition of $\perp$ lines. 4. Definition of a right triangle.
L 5. $\overline{EG}=\overline{EG}$ 6. $△CEG\cong △FEG$	5. Reflexive Property of Congruence 6. HL Congruence Theorem

Now,

There are four triangles $△ABD,△CBD,△CEG\text{ and }△FEG\text{.}$

Statements	Reasons
H 1. $\overline{AB}\cong \overline{CB}\cong \overline{CE}\cong \overline{FE}.$	1.Given
L 5. $\overline{BD}\cong \overline{EG}$	5. Reflexive Property of Congruence

Also, all the triangles are right triangles .Therefore, by HL Congruence Theorem all the four triangles are congruent.

  $$

  $△ABD\cong △CBD\cong △CEG\cong △FEG\text{.}$

Hence proved.