Question

similar triangles
study the example showing how to use angle measures to identify similar triangles. then solve problems 1 - 6.
example
which triangles are similar?
3x + 5x = 160
8x = 160
x = 20
m∠hgj = 60°, m∠ghj = 100°
m∠gjh = 180° - 160° = 20°
100 + y + y - 20 = 180
2y + 80 = 180
2y = 100
y = 50
m∠lkm = 50°, m∠klm = 30°
m∠srt = 180° - 120° = 60°
m∠rts = 20°
∠hgj ≅ ∠srt and ∠gjh ≅ ∠rts, so △ghj ~ △rst.
1 is △abc similar to any of the triangles in the example? explain.
2 triangle x has two angles that measure 80° and 30°. triangle y has two angles that measure 80° and 70°. hannah says that triangles x and y are not similar. jasmine says they are similar. who is correct? explain.
lesson 7 describe angle relationships in triangles
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Explanation:

Step1: Find angles of $\triangle ABC$

Use the exterior - angle property. The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, $(2x - 10)=x+(x + 50)$. Simplifying gives $2x-10 = 2x + 50$, which is a contradiction. Let's use the angle - sum property of a triangle. The sum of the interior angles of a triangle is $180^{\circ}$. So, $x+(180-(2x - 10))+(180-(x + 50))=180$. First, expand: $x + 180-2x + 10+180 - x - 50 = 180$. Combine like terms: $(x-2x - x)+(180 + 10+180 - 50)=180$, $-2x+320 = 180$. Then, $-2x=180 - 320=-140$, and $x = 70$. So, $m\angle ABC=70^{\circ}$, $m\angle BAC=180-(2\times70 - 10)=50^{\circ}$, $m\angle ACB=180-(70 + 50)=60^{\circ}$.

Step2: Compare with example triangles

In the example, $\triangle GHJ$ has angles $60^{\circ},100^{\circ},20^{\circ}$; $\triangle LKM$ has angles $50^{\circ},30^{\circ},100^{\circ}$; $\triangle RST$ has angles $60^{\circ},20^{\circ},100^{\circ}$. Since the angle measures of $\triangle ABC$ ($50^{\circ},60^{\circ},70^{\circ}$) do not match the angle measures of any of the triangles in the example, $\triangle ABC$ is not similar to any of them.

Step3: Analyze triangles $X$ and $Y$

For triangle $X$ with angles $80^{\circ}$ and $70^{\circ}$, the third angle is $180-(80 + 70)=30^{\circ}$. For triangle $Y$ with angles $80^{\circ}$ and $30^{\circ}$, the third angle is $180-(80 + 30)=70^{\circ}$. Since the angle measures of triangle $X$ and triangle $Y$ are the same, the two triangles are similar by the AA (angle - angle) similarity criterion. So Jasmine is correct.

Answer:

1. $\triangle ABC$ is not similar to any of the triangles in the example because its angle measures ($50^{\circ},60^{\circ},70^{\circ}$) do not match the angle measures of $\triangle GHJ$, $\triangle LKM$, or $\triangle RST$.
2. Jasmine is correct. Triangle $X$ has angles $80^{\circ},70^{\circ},30^{\circ}$ and triangle $Y$ has angles $80^{\circ},30^{\circ},70^{\circ}$. By the AA similarity criterion, they are similar.