Question

unit 4 quiz 3
willhite_k. geometry a_section 1_fall 25/26 / unit 4 - introduction to polygons and triangles
given: right \\( \triangle a b d \\) and right \\( \triangle c b d \\)
prove: \\( \triangle a b d \cong \triangle c b d \\)
proof:
find the length of \\( \overline{a d} \\)
\\( \

$$\begin{aligned}a d & =\\sqrt{\\left(x_2-x_1\ ight)^2+\\left(y_2-y_1\ ight)^2} \\\\& =\\sqrt{(4-0)^2+(0-0)^2} \\\\& =\\sqrt{4^2}=\\end{aligned}$$

\\)
find the length of \\( \overline{d c} \\)
\\( \

$$\begin{aligned}d c & =\\sqrt{\\left(x_2-x_1\ ight)^2+\\left(y_2-y_1\ ight)^2} \\\\& =\\sqrt{(8-4)^2+(0-0)^2} \\\\& =\\sqrt{4^2}=\\end{aligned}$$

\\)

• \\( \overline{a d} \cong \overline{d c} \\) by the
• \\( \overline{b d} \cong \overline{b d} \\) by the
• \\( \triangle a b d \cong \triangle c b d \\) by the congruence theorem

a. 4
b. reflexive property of \\( \cong \\)
c. segment property
d. 8
e. 16
f. leg leg
g. definition of \\( \cong \\)

Explanation:

Step1: Calculate the length of \(AD\)

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Step2: Calculate the length of \(DC\)

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Since \(AD = 4\) and \(DC=4\), by the definition of congruent segments (\(AD\cong DC\) when \(AD = DC\)).

Step3: Justify \(BD\cong BD\)

\(BD\cong BD\) by the Reflexive Property of \(\cong\) (a segment is congruent to itself).

Step4: Determine the congruence theorem

Since we have two right - triangles (\(\triangle ABD\) and \(\triangle CBD\)), and we have shown that \(AD\cong DC\) (legs) and \(BD\cong BD\) (common leg), \(\triangle ABD\cong\triangle CBD\) by the Leg - Leg (\(LL\)) Congruence Theorem for right - triangles.

Answer:

For the length of \(AD\) and \(DC\): \(4\) (option a). \(\overline{AD}\cong\overline{DC}\) by the definition of \(\cong\) (option g). \(\overline{BD}\cong\overline{BD}\) by the Reflexive Property of \(\cong\) (option b). \(\triangle ABD\cong\triangle CBD\) by the Leg - Leg (\(LL\)) Congruence Theorem (option f).