Question

self - assessment 1 i dont understand yet. 2 i can do it with help. 3 i can do it on my own. 4 i can teach someone else.

1. in the diagram, \\( \overline { a b } \perp \overline { a d } , \overline { d e } \perp \overline { a d } \\), and \\( \overline { a c } \cong \overline { d c } \\). prove \\( \triangle a b c \cong \triangle d e c \\).

in the diagram, \\( \angle s \cong \angle u \\) and \\( \overline { r s } \cong \overline { v u } \\). prove \\( \triangle r s t \cong \triangle v u t \\).

Explanation:

Step1: Analyze the first problem ($\triangle ABC\cong\triangle DEC$)

• Statements:
• $\overline{AB}\perp\overline{AD},\overline{DE}\perp\overline{AD}$
• $\angle A=\angle D = 90^{\circ}$
• $\overline{AC}\cong\overline{DC}$
• $\angle ACB=\angle DCE$ (vertical angles)
• Reasons:
• Given
• Definition of perpendicular lines
• Given
• Vertical angles are congruent
• By the Angle - Angle - Side (AAS) congruence criterion, $\triangle ABC\cong\triangle DEC$

Step2: Analyze the second problem ($\triangle RST\cong\triangle VUT$)

• Statements:
• $\angle S\cong\angle U$
• $\overline{RS}\cong\overline{VU}$
• $\angle RTS=\angle VTU$ (vertical angles)
• Reasons:
• Given
• Given
• Vertical angles are congruent
• By the Angle - Angle - Side (AAS) congruence criterion, $\triangle RST\cong\triangle VUT$

Answer:

For the first problem ($\triangle ABC\cong\triangle DEC$):

STATEMENTS	REASONS
$\angle A=\angle D = 90^{\circ}$	Definition of perpendicular lines
$\overline{AC}\cong\overline{DC}$	Given
$\angle ACB=\angle DCE$	Vertical angles are congruent
$\triangle ABC\cong\triangle DEC$	AAS congruence criterion

For the second problem ($\triangle RST\cong\triangle VUT$):

STATEMENTS	REASONS
$\overline{RS}\cong\overline{VU}$	Given
$\angle RTS=\angle VTU$	Vertical angles are congruent
$\triangle RST\cong\triangle VUT$	AAS congruence criterion