Question

guided notes lesson 8 - 3: aa similarity
i can prove triangle similarity using angle - angle. (ma.912.gr.1.2)
i can solve problems involving similarity in triangles. (ma.912.gr.1.6)
learn:
postulate 8.1: angle - angle (aa) similarity
if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
example: if ∠a ≅ ∠y and ∠b ≅ ∠z, then δabc ≅ δyzx.
when solving for unknown quantities in similar triangles, solving proportions will be utilized. see below for a review from algebra 1.
\\(\frac{x}{9}=\frac{2x - 3}{24}\\)
24(x) = 9(2x - 3) cross multiply
24x = 18x - 27 distribute
6x = - 27 subtract 18x from both sides
x = - 4.5 divide both sides by 6
practice solving proportions:

1. \\(\frac{t}{0.3}=\frac{17}{0.9}\\)
2. \\(\frac{x}{3}=\frac{3x + 2}{6}\\)
3. \\(\frac{5}{7k + 4}=\frac{2}{2k - 3}\\)

example 1: determine whether the triangles are similar. explain your reasoning.
∠l ≅ ∠l by the \underline{\qquad} property of congruence.
∠lpq ≅ \underline{\qquad} by the corresponding angles theorem.
by aa similarity, \underline{\qquad}
example 1 check: determine whether the triangles are similar. explain your reasoning.
by the triangle angle - sum theorem, m∠a = \underline{\qquad} ° and m∠g = \underline{\qquad} °
because \underline{\qquad} two pairs of angles are congruent,
△abc is / is not similar to △fgd. (circle the correct choice)

Explanation:

Practice Solving Proportions

Step1: Cross multiply to eliminate fractions

$0.9t = 0.3 \times 17$

Step2: Calculate right-hand side value

$0.9t = 5.1$

Step3: Solve for t by dividing

$t = \frac{5.1}{0.9} = \frac{17}{3} \approx 5.67$

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Step1: Cross multiply to eliminate fractions

$6x = 3(3x+2)$

Step2: Distribute the right-hand side

$6x = 9x + 6$

Step3: Isolate x terms

$6x - 9x = 6$

Step4: Solve for x

$-3x = 6 \implies x = -2$

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Step1: Cross multiply to eliminate fractions

$5(2k-3) = 2(7k+4)$

Step2: Distribute both sides

$10k - 15 = 14k + 8$

Step3: Isolate k terms

$10k - 14k = 8 + 15$

Step4: Solve for k

$-4k = 23 \implies k = -\frac{23}{4} = -5.75$

• $\angle L$ is a shared angle between the two triangles, so it is congruent to itself via the Reflexive Property.
• Segments $PQ$ and $JK$ are parallel, so corresponding angles formed by transversal $LK$ are congruent: $\angle LPQ \cong \angle LJK$.
• With two pairs of congruent angles, the triangles satisfy AA Similarity.

Step1: Calculate $m\angle A$ using angle sum

$m\angle A = 180^\circ - 90^\circ - 44^\circ = 46^\circ$

Step2: Calculate $m\angle G$ using angle sum

$m\angle G = 180^\circ - 90^\circ - 47^\circ = 43^\circ$

Step3: Compare angle pairs

The right angles are congruent, but the other angle pairs ($46^\circ
eq 47^\circ$, $44^\circ
eq 43^\circ$) are not congruent, so no two pairs of congruent angles exist.

Answer:

1. $t = \frac{17}{3}$
2. $x = -2$
3. $k = -\frac{23}{4}$

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Example 1