Question

1. what additional information is needed to prove \\( \triangle j k l \cong \triangle j m l \\) by the sas postulate?

a. \\( \angle j l k \cong \angle j l m \\)
b. \\( \underline{j k} \cong \underline{j m} \\)
c. \\( \angle j k l \cong \angle j m l \\)
d. \\( \underline{k l} \cong \underline{m l} \\)

1. classify the following triangle by side lengths.

f. equilateral
g. scalene
h. isosceles
j. equiangular

1. \\( \triangle p q r \\) is isosceles, with \\( \underline{p q} \cong \underline{r q} \\). which is the length of \\( \underline{p r} \\)?

a. \\( \underline{p r}=6 \\)
b. \\( \underline{p r}=11 \\)
c. \\( \underline{p r}=14 \\)
d. \\( \underline{p r}=18 \\)

Explanation:

Question 5

Step1: Recall SAS postulate

SAS (Side - Angle - Side) postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In \(\triangle JKL\) and \(\triangle JML\), we already know \(LK = LM\) (marked with the same tick - mark) and \(LJ\) is common (\(LJ = LJ\)).

Step2: Check each option

• Option A: \(\angle JLK\) and \(\angle JLM\) are not the included angles for the sides \(LK,LJ\) and \(LM,LJ\).
• Option B: \(JK\) and \(JM\) are not the sides that we are using for the SAS postulate (we already have \(LK = LM\) and \(LJ\) common).
• Option C: \(\angle JKL\) and \(\angle JML\) are not the included angles.
• Option D: If \(KL\cong ML\) (we already have \(LK = LM\) from the figure's marking, assume it's a typo and they mean \(KL = ML\) as given, and \(\angle KLJ=\angle MLJ\) (the included angle as \(LJ\) is common), then by SAS \(\triangle JKL\cong\triangle JML\).

Step1: Recall triangle side - length classification

• An equilateral triangle has all three sides equal.
• A scalene triangle has all three sides of different lengths.
• An isosceles triangle has at least two sides equal.
• An equiangular triangle (a special case of equilateral) has all angles equal (and all sides equal).

Step2: Check the side lengths

The side lengths of the triangle are \(9\), \(13\), and \(14\). Since \(9
eq13
eq14\), all three sides have different lengths.

Step1: Use the property of isosceles triangle (\(PQ = RQ\))

Since \(\triangle PQR\) is isosceles with \(PQ\cong RQ\), we set \(3x - 6=8x - 26\).

Step2: Solve for \(x\)

$$
LATEXBLOCK0
$$

Step3: Find the length of \(PR\)

Substitute \(x = 4\) into \(PR=2x + 3\). Then \(PR=2\times4+3=8 + 3=11\).

Answer:

D. \(KL\cong ML\)

Question 6