Question

1. draw two triangles that are similar by the aa similarity theorem but not congruent. each triangle can only identify 2 angles.
2. draw two triangles that are not similar by the aa similarity theorem. each triangle can only identify 2 angles.
3. draw two triangles that are similar and congruent. no restriction on sides, but only include information on two angle pairs.

draw the triangle that is similar to triangle cat but a scale factor k=1.5.

Explanation:

Step1: Calculate scaled side lengths

Multiply each side by $k=1.5$:
$\text{Side } C'A' = 18 \times 1.5 = 27$
$\text{Side } A'T' = 13 \times 1.5 = 19.5$
$\text{Side } T'C' = 7 \times 1.5 = 10.5$

Step2: Draw similar triangle

Construct $\triangle C'A'T'$ with angles congruent to $\triangle CAT$, and sides equal to the calculated scaled lengths.

Step1: Define first triangle angles

Choose $\triangle ABC$ with $\angle A=60^\circ$, $\angle B=50^\circ$

Step2: Define second similar triangle

Choose $\triangle DEF$ with $\angle D=60^\circ$, $\angle E=50^\circ$, sides 2x length of $\triangle ABC$

Step3: Verify non-congruence

Sides are proportional, not equal, so no congruence.

Step1: Define first triangle angles

$\triangle GHI$ with $\angle G=30^\circ$, $\angle H=60^\circ$

Step2: Define non-similar triangle

$\triangle JKL$ with $\angle J=30^\circ$, $\angle K=70^\circ$

Step3: Verify non-similarity

Only one matching angle, so AA fails.

Step1: Define first triangle angles

$\triangle MNO$ with $\angle M=90^\circ$, $\angle N=45^\circ$

Step2: Define congruent triangle

$\triangle PQR$ with $\angle P=90^\circ$, $\angle Q=45^\circ$, identical side lengths

Step3: Verify similarity & congruence

Matching angles (AA similarity) and equal sides (congruence).

Answer:

The similar triangle $\triangle C'A'T'$ has side lengths: $27$, $19.5$, and $10.5$, with all corresponding angles equal to those in $\triangle CAT$.

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For Question 5 (partial text):