Question

triangle congruence proofs
class work
write a two - column or flow proof.
143.
given:
qt ≅ st
tr bisects qs
prove: △qrt ≅ △srt
(image of triangle qts with r on qs and t below qs)
144.
given:
mp ≅ op
np bisects ∠mpo
prove: △pmn ≅ △pon
(image of quadrilateral mnop with diagonal np)
145.
given:
wx ⊥ vy, zu ⊥ vy
uz ≅ xw, ∠x ≅ ∠u
prove: ∠y ≅ ∠v
(image of quadrilateral yvux with z on yu and w on vx)
146.
given:
∠i & ∠k are right ∠s
j is mid pt. of hl
prove: △ihj ≅ △klj
(image of two right triangles ihj and klj with j on ik and hl)
home work
write a two - column or flow proof.
147.
given:
gh ≅ ds
hd bisects sg
sg bisects hd
prove: △fds ≅ △fgh
(image of two triangles fds and fgh with f as the common vertex)
148.
given:
line r || line p
fg ≅ fi
prove: △ghf ≅ △ihf
(image of triangles ghf and ihf with lines p, r, q and points f, g, i)
149.
given:
om ⊥ pu
op ≅ ou
prove: △pmo ≅ △umo
(image of isosceles triangle pou with m on pu and om perpendicular to pu)
150.
given:
qo ⊥ np
nq ≅ pt
prove: ∠noq ≅ ∠poq
(image of triangle nop with q on np and oq perpendicular to np)

Explanation:

Step1: Analyze Given Information

Given \( \overline{QT} \cong \overline{ST} \), \( TR \) bisects \( \overline{QS} \), so \( \overline{QR} \cong \overline{SR} \) (definition of bisector). Also, \( \overline{TR} \) is common to both \( \triangle QRT \) and \( \triangle SRT \).

Step2: Apply SSS Congruence Criterion

In \( \triangle QRT \) and \( \triangle SRT \):

• \( \overline{QT} \cong \overline{ST} \) (Given)
• \( \overline{QR} \cong \overline{SR} \) (TR bisects QS)
• \( \overline{TR} \cong \overline{TR} \) (Reflexive property)

By SSS (Side - Side - Side) congruence postulate, \( \triangle QRT \cong \triangle SRT \).

Answer:

\( \triangle QRT \cong \triangle SRT \) (proven by SSS congruence using given \( \overline{QT} \cong \overline{ST} \), \( \overline{QR} \cong \overline{SR} \) (from TR bisecting QS), and common \( \overline{TR} \))