4. given $ \triangle stu \cong \triangle klm $, complete each of the following statements.
a) $ \overline{tu} \cong $______ d) $ \angle m \cong $______ g) $ \triangle ust \cong $______
b) $ \overline{km} \cong $______ e) $ \angle t \cong $______ h) $ \triangle tus \cong $______
c) $ \overline{lk} \cong $______ f) $ \angle ust \cong $______
5. given $ \triangle bcm \cong \triangle zyr $, find each missing measure.
a) $ cm= $______ d) $ m\angle b= $______
b) $ bm= $______ e) $ m\angle m= $______
c) $ yz= $______ f) $ m\angle y= $______
6. given $ \triangle acp \cong \triangle lnx $, find each missing measure.
a) $ xl= $______ d) $ m\angle l= $______
b) $ ac= $______ e) $ m\angle c= $______
c) $ pc= $______ f) $ m\angle x= $______

Question

Accepted Answer

# Explanation:
## Step1: Use the property of congruent triangles
When two triangles are congruent, their corresponding sides and angles are equal.
For $\triangle STU\cong\triangle KLM$:
- Corresponding sides: $ST = KL$, $TU=LM$, $US = MK$
- Corresponding angles: $\angle S=\angle K$, $\angle T=\angle L$, $\angle U=\angle M$

## Step2: Answer part (a) - (h)
a) Since $TU$ corresponds to $LM$ in $\triangle KLM$, $\overline{TU}\cong\overline{LM}$
b) Since $KM$ corresponds to $US$ in $\triangle STU$, $\overline{KM}\cong\overline{US}$
c) Since $LK$ corresponds to $ST$ in $\triangle STU$, $\overline{LK}\cong\overline{ST}$
d) Since $\angle M$ corresponds to $\angle U$ in $\triangle STU$, $\angle M\cong\angle U$
e) Since $\angle T$ corresponds to $\angle L$ in $\triangle KLM$, $\angle T\cong\angle L$
f) $\angle UST$ corresponds to $\angle MKL$ in $\triangle KLM$, $\angle UST\cong\angle MKL$
g) $\triangle UST\cong\triangle MKL$ (by SSS or SAS congruence as corresponding sides and angles are equal)
h) $\triangle TUS\cong\triangle LMK$ (by SSS or SAS congruence as corresponding sides and angles are equal)

## Step3: For $\triangle BCM\cong\triangle ZYR$
- Corresponding sides: $BC = ZY$, $CM = YR$, $BM = ZR$
- Corresponding angles: $\angle B=\angle Z$, $\angle C=\angle Y$, $\angle M=\angle R$
a) $CM = YR$. From the figure $YR = 11m$, so $CM = 11m$
b) $BM = ZR$. From the figure $ZR = 15m$, so $BM = 15m$
c) $YZ = BC$. From the figure $BC = 8m$, so $YZ = 8m$
d) $m\angle B=m\angle Z$. From the figure $m\angle Z = 45^{\circ}$, so $m\angle B = 45^{\circ}$
e) $m\angle M=m\angle R$. Since the sum of angles in a triangle is $180^{\circ}$, $m\angle R=180-(103 + 45)=32^{\circ}$, so $m\angle M = 32^{\circ}$
f) $m\angle Y=m\angle C$. From the figure $m\angle C = 103^{\circ}$, so $m\angle Y = 103^{\circ}$

## Step4: For $\triangle ACP\cong\triangle LNX$
- Corresponding sides: $AC = LN$, $CP = NX$, $AP = LX$
- Corresponding angles: $\angle A=\angle L$, $\angle C=\angle N$, $\angle P=\angle X$
a) $XL = AP$. From the figure $AP = 13cm$, so $XL = 13cm$
b) $AC = LN$. Since $LN = 21cm$ (from the figure as $LN$ is equal to the side marked $21cm$), $AC = 21cm$
c) $PC = NX$ (by congruence)
d) $m\angle L=m\angle A$. From the figure $m\angle A = 29^{\circ}$, so $m\angle L = 29^{\circ}$
e) $m\angle C=m\angle N$. Since the sum of angles in a triangle is $180^{\circ}$, $m\angle N=180-(29 + 90)=61^{\circ}$ (assuming $\triangle ACP$ has $\angle P = 90^{\circ}$ as $\triangle LNX$ is isoceles (from the marks on sides) and using angle - sum property), $m\angle C = 61^{\circ}$
f) $m\angle X=m\angle P$. Since $\angle P = 90^{\circ}$ (by assuming the right - angle from the side - length marks and triangle congruence), $m\angle X = 90^{\circ}$

# Answer:
a) $\overline{LM}$
b) $\overline{US}$
c) $\overline{ST}$
d) $\angle U$
e) $\angle L$
f) $\angle MKL$
g) $\triangle MKL$
h) $\triangle LMK$

For $\triangle BCM\cong\triangle ZYR$:
a) $11m$
b) $15m$
c) $8m$
d) $45^{\circ}$
e) $32^{\circ}$
f) $103^{\circ}$

For $\triangle ACP\cong\triangle LNX$:
a) $13cm$
b) $21cm$
c) $NX$
d) $29^{\circ}$
e) $61^{\circ}$
f) $90^{\circ}$

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QUESTION IMAGE

Question

Explanation:

Step1: Use the property of congruent triangles

Step2: Answer part (a) - (h)

Step3: For \(\triangle BCM\cong\triangle ZYR\)

Step4: For \(\triangle ACP\cong\triangle LNX\)

Answer: