Question

12.△qrs≅?
f.△zyx
g.△yzx
h.△xzy
j.△yxz

1. what congruence postulate could be used to prove △qrs≅△qrt?

a. sas b. asa c. sss d. ssa

1. given:△abc≅△lkm

solve for x.

Explanation:

12.

When triangles are congruent, their corresponding vertices are in order. In \(\triangle QRS\) and \(\triangle YXZ\), the sides and angles correspond. By matching the congruent parts (marked with equal - side and equal - angle symbols), we can see that \(Q\) corresponds to \(Y\), \(R\) corresponds to \(X\), and \(S\) corresponds to \(Z\).

• For \(\triangle QRS\) and \(\triangle QRT\):
• \(QR = QR\) (common side).
• \(QS=QT\) (given, if we assume from the figure structure, in a congruence - proof context for these two triangles, we have two sides and the included angle.
• \(\angle SQR=\angle TQR\) (given, from the angle - marking).
• The \(SAS\) (Side - Angle - Side) postulate states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Step 1: Find the measure of \(\angle C\) in \(\triangle ABC\)

In \(\triangle ABC\), using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)).
We know \(\angle A = 20^{\circ}\) and \(\angle B=86^{\circ}\).

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Step 2: Use the congruence of \(\triangle ABC\) and \(\triangle LKM\)

Since \(\triangle ABC\cong\triangle LKM\), \(\angle C=\angle L\) (corresponding angles of congruent triangles are equal).
We are given \(\angle L=(9x + 2)^{\circ}\) and \(\angle C = 74^{\circ}\).
Set up the equation \(9x+2=74\).
Subtract 2 from both sides: \(9x=74 - 2=72\).
Divide both sides by 9: \(x=\frac{72}{9}=8\).

Answer:

J. \(\triangle YXZ\)

13.