Question

given: rhombus tcbv,
\\( \overline { c u } \perp \overline { b v } \\), \\( \overline { c s } \perp \overline { t v } \\)
prove: \\( \overline { c u } \cong \overline { c s } \\)

1. \\( \angle t \cong \angle b \\)
2. corresponding parts

of congruent tri-
angles are congruent
(c.p.c.t.c.)

1. \\( \triangle c t s \cong \triangle c b u \\)
2. angle - angle - side
3. click here to insert
4. click here to insert

Explanation:

Step1: Recall properties of a rhombus

All sides of a rhombus are equal. So, \(CT = CB\) in rhombus \(TCBV\).

Step2: Use right - angle property

Since \(\overline{CU}\perp\overline{BV}\) and \(\overline{CS}\perp\overline{TV}\), \(\angle CSU=\angle CUB = 90^{\circ}\)

Step3: Use AAS congruence

We have \(\angle T\cong\angle B\) (given), \(\angle CSU=\angle CUB\) (right angles), and \(CT = CB\) (sides of a rhombus). By Angle - Angle - Side (AAS) congruence criterion, \(\triangle CTS\cong\triangle CBU\)

Step4: Apply CPCTC

If \(\triangle CTS\cong\triangle CBU\), then by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), \(\overline{CU}\cong\overline{CS}\)

Answer:

1. \(\overline{CU}\cong\overline{CS}\) ; 7. Corresponding Parts of Congruent Triangles are Congruent (C.P.C.T.C.)