Question

4-4 additional practice
proving and applying the asa and aas congruence criteria
for exercises 1 and 2, complete each two - column proof.

1. given: (overline{ad}) bisects (angle a).

prove: (\triangle adbcong\triangle adc)

1. given: (angle pcongangle s), (overline{tq}congoverline{rq})

prove: (\triangle qrscong\triangle qtp)

1. understand chen says there is not enough information to prove that (\triangle abccong\triangle cda). explain why chen is incorrect.

1. apply casey builds the gate shown, with two solid triangular panels. each support post runs 10 cm above and below the gate. are the solid panels congruent? explain.

Explanation:

(Exercise 1):

Step1: State given reason

1. Given

Step2: Define angle bisector result

$\angle CAD \cong \angle BAD$

1. Definition of Angle Bisector

Step3: State reflexive congruence

$\overline{AD} \cong \overline{AD}$

1. Reflexive Property of Congruence

Step4: Define right angles reason

1. Definition of Perpendicular Lines

Step5: State ASA congruence rule

1. ASA Congruence Postulate

(Exercise 2):

Step1: List given statements

1. $\angle P \cong \angle S$, $\overline{TQ} \cong \overline{RQ}$

Step2: State vertical angles reason

1. Vertical Angles are Congruent

Step3: State final congruence statement

$\triangle QRS \cong \triangle QTP$

(Exercise 3):

Step1: Identify given info

$\angle B$ and $\angle D$ are right angles, so $\angle B \cong \angle D$; $\overline{AC} \cong \overline{CA}$ (reflexive); $\overline{BC} \cong \overline{AD}$ (opposite sides of rectangle are congruent)

Step2: Apply congruence rule

We can use HL (right triangles) or AAS to prove $\triangle ABC \cong \triangle CDA$. Chen is incorrect because there is sufficient information.

(Exercise 4):

Step1: Identify congruent parts

Both panels have a 2.5 m side, a shared (congruent) base side, and a $38^\circ$ angle between these sides.

Step2: Apply SAS congruence

By SAS Congruence Postulate, the triangular panels are congruent.

Answer:

1.

Statements	Reasons
$\angle CAD \cong \angle BAD$	2. Definition of Angle Bisector
$\overline{AD} \cong \overline{AD}$	3. Reflexive Property of Congruence
$\angle CDA$ and $\angle BDA$ are right angles.	4. Definition of Perpendicular Lines
$\angle CDA \cong \angle BDA$	5. All right angles are congruent.
$\triangle ADB \cong \triangle ADC$	6. ASA Congruence Postulate

2.

Statements	Reasons
$\angle RQS \cong \angle TQP$	2. Vertical Angles are Congruent
$\triangle QRS \cong \triangle QTP$	3. AAS

1. Chen is incorrect because: $\triangle ABC$ and $\triangle CDA$ are right triangles, $\overline{AC} \cong \overline{CA}$ (reflexive), $\overline{BC} \cong \overline{AD}$ (opposite sides of a rectangle are congruent). We can use the HL Congruence Theorem (for right triangles) or AAS to prove $\triangle ABC \cong \triangle CDA$, so there is enough information.

1. Yes, the solid panels are congruent. Each panel has a 2.5 m side, a congruent base side (shared between the two panels), and a $38^\circ$ angle between these two sides. By the SAS Congruence Postulate, the two triangular panels are congruent.