Question

8 multiple choice 1 point in the figure, \\( \overline{nq} \\) is parallel to \\( \overline{op} \\) and \\( nq = 4 \\), \\( op = 6 \\), and \\( mq = 8 \\). if \\( no = 2 \\), how long is \\( \overline{qp} \\)? 9 multiple choice 1 point if two scalene triangles have two congruent angles, then the triangles must be— similar acute congruent right

Explanation:

Step1: Use the similarity of triangles

Since \(\overline{NQ}\parallel\overline{OP}\), \(\triangle MNQ\sim\triangle MOP\) (by the AA similarity criterion, as corresponding angles are equal).

Step2: Set up the proportion

For similar triangles \(\triangle MNQ\) and \(\triangle MOP\), we have the proportion \(\frac{NQ}{OP}=\frac{MQ}{MP}\). Let \(QP = x\), then \(MP=MQ + QP=8 + x\).
We know \(NQ = 4\), \(OP = 6\), \(MQ = 8\). Substituting into the proportion: \(\frac{4}{6}=\frac{8}{8 + x}\).

Step3: Cross - multiply

Cross - multiplying gives \(4(8 + x)=6\times8\).
Expanding the left - hand side: \(32+4x = 48\).

Step4: Solve for \(x\)

Subtract 32 from both sides: \(4x=48 - 32\), so \(4x = 16\).
Divide both sides by 4: \(x = 4\).

for question 9:
If two scalene triangles have two congruent angles, then by the AA (Angle - Angle) similarity criterion, the third angle is also congruent (since the sum of angles in a triangle is \(180^{\circ}\)). Similar triangles have the same shape (corresponding angles are equal) but not necessarily the same size (corresponding sides are in proportion). Congruent triangles require all corresponding sides and angles to be equal. Acute triangles are defined by their angles being less than \(90^{\circ}\) and right triangles have one \(90^{\circ}\) angle. Since we only know about two congruent angles, the triangles must be similar.

Answer:

4