Question

triangle congruence proofs
class work
write a two - column or flow proof.
143.
given:
$overline{qt} \cong \overline{st}$
$overline{tr}$ bisects $overline{qs}$
prove: $\triangle qrt \cong \triangle srt$
(accompanying triangle image with vertices q, r, s, t)
144.
given:
$overline{mp} \cong \overline{op}$
$overline{np}$ bisects $\angle mpo$
prove: $\triangle pmn \cong \triangle pon$
(accompanying quadrilateral - triangle image with vertices m, n, o, p)
145.
given:
$overline{wx} \perp \overline{vy}$, $overline{zu} \perp \overline{vy}$
$overline{uz} \cong \overline{xw}$, $\angle x \cong \angle u$
prove: $\angle y \cong \angle v$
(accompanying polygon image with vertices u, v, x, w, z, y)
146.
given:
$\angle i$ & $\angle k$ are right $\angle$s
j is mid pt. of $overline{hl}$
prove: $\triangle ihj \cong \triangle klj$
(accompanying triangle - segment image with vertices h, i, j, k, l)

Explanation:

Problem 143:

Step 1: Identify Given Information

We are given \( QT \cong ST \) and \( TR \) bisects \( QS \). By the definition of a bisector, \( QR \cong SR \).

Step 2: Identify Common Side

The side \( TR \) is common to both \( \triangle QRT \) and \( \triangle SRT \), so \( TR \cong TR \) (Reflexive Property of Congruence).

Step 3: Apply SSS Congruence Criterion

We have \( QT \cong ST \), \( QR \cong SR \), and \( TR \cong TR \). By the Side - Side - Side (SSS) Congruence Postulate, \( \triangle QRT \cong \triangle SRT \).

Step 1: Identify Given Information

We are given \( MP \cong OP \) and \( NP \) bisects \( \angle MPO \), so \( \angle MPN \cong \angle OPN \) (Definition of Angle Bisector). Also, \( NP \cong NP \) (Reflexive Property of Congruence).

Step 2: Apply SAS Congruence Criterion

We have \( MP \cong OP \), \( \angle MPN \cong \angle OPN \), and \( NP \cong NP \). By the Side - Angle - Side (SAS) Congruence Postulate, \( \triangle PMN \cong \triangle PON \).

Step 1: Identify Right Angles

Since \( WX\perp VY \) and \( ZU\perp VY \), \( \angle WXV = \angle UZV=90^{\circ} \) (Definition of Perpendicular Lines).

Step 2: Identify Given Congruences

We are given \( UZ \cong XW \) and \( \angle X\cong\angle U \).

Step 3: Apply ASA or AAS Congruence Criterion

In \( \triangle XWV \) and \( \triangle UZV \), we have \( \angle WXV=\angle UZV \), \( UZ \cong XW \), and \( \angle X\cong\angle U \). By the Angle - Side - Angle (ASA) or Angle - Angle - Side (AAS) Congruence Postulate, \( \triangle XWV \cong \triangle UZV \). Then, corresponding angles \( \angle Y \) and \( \angle V \) are congruent.

Answer:

Using SSS congruence ( \( QT \cong ST \), \( QR \cong SR \), \( TR \cong TR \) ), we prove \( \triangle QRT \cong \triangle SRT \).

Problem 144: