unit: angle relationships
student handout 5
name
date
pd
angles in similar triangles
similar figures are the same but may not necessarily be the same size. similar
figures have sides, meaning the ratios of corresponding sides are .
is triangle abc similar to triangle def?
explain.
is triangle abc similar to triangle fgh?
explain.
what do you notice about the angle measures of the similar triangles above?
the angle - angle criterion described below allows us to determine if two triangles are similar by
observing just their angle measures.
angle - angle
criterion
- if two pairs of corresponding angles in a pair of triangles are
, then the two triangles are .
- rewrite the angle - angle criterion in your own words:
in 1 - 3, use the angle - angle criterion to determine if the triangles are similar. explain your choice.
1.
2.
3. jace constructed triangle abc, where ( mangle a = 53^{circ} ), and ( mangle b = 112^{circ} ). ruthie constructed
triangle rst, where ( mangle s = 112^{circ} ), and ( mangle t = 15^{circ} ). are the two triangles similar? explain.
©maneuvering the middle llc, 2017

Question

unit: angle relationships
student handout 5
name
date
pd
angles in similar triangles
similar figures are the same  but may not necessarily be the same size. similar
figures have  sides, meaning the ratios of corresponding sides are .
is triangle abc similar to triangle def?
explain.
is triangle abc similar to triangle fgh?
explain.
what do you notice about the angle measures of the similar triangles above?
the angle - angle criterion described below allows us to determine if two triangles are similar by
observing just their angle measures.
angle - angle
criterion
- if two pairs of corresponding angles in a pair of triangles are
, then the two triangles are .
- rewrite the angle - angle criterion in your own words:
in 1 - 3, use the angle - angle criterion to determine if the triangles are similar. explain your choice.
1.
2.
3. jace constructed triangle abc, where ( mangle a = 53^{circ} ), and ( mangle b = 112^{circ} ). ruthie constructed
triangle rst, where ( mangle s = 112^{circ} ), and ( mangle t = 15^{circ} ). are the two triangles similar? explain.
©maneuvering the middle llc, 2017

Accepted Answer

### Part 1: Filling in the Blanks about Similar Figures
# Explanation:
## Step1: Recall the definition of similar figures. Similar figures have the same shape (they look alike in terms of their angles and the proportions of their sides) but can be different sizes. So the first blank is "shape".
## Step2: Similar figures have proportional sides. This means that the ratios of corresponding sides are equal. So the second blank is "proportional" and the third blank is "equal".
# Answer:
Similar figures are the same $\boldsymbol{\text{shape}}$ but may not necessarily be the same size. Similar figures have $\boldsymbol{\text{proportional}}$ sides, meaning the ratios of corresponding sides are $\boldsymbol{\text{equal}}$.

### Part 2: Determining Similarity of Triangles (ABC vs DEF)
# Explanation:
## Step1: Find the angles of triangle ABC. In triangle ABC, $\angle A = 30^\circ$, $\angle B = 60^\circ$, so $\angle C=180 - 30 - 60=90^\circ$ (since the sum of angles in a triangle is $180^\circ$).
## Step2: Find the angles of triangle DEF. In triangle DEF, $\angle D = 35^\circ$, $\angle E = 65^\circ$, so $\angle F = 180 - 35 - 65 = 80^\circ$.
## Step3: Compare the angles. The angles of ABC are $30^\circ$, $60^\circ$, $90^\circ$ and angles of DEF are $35^\circ$, $65^\circ$, $80^\circ$. No two pairs of angles are equal. So triangle ABC is not similar to triangle DEF.
# Answer:
Triangle ABC is not similar to triangle DEF. In $\triangle ABC$, angles are $30^\circ$, $60^\circ$, $90^\circ$. In $\triangle DEF$, angles are $35^\circ$, $65^\circ$, $80^\circ$. No two pairs of corresponding angles are equal, so by the angle - angle criterion, they are not similar.

### Part 3: Determining Similarity of Triangles (ABC vs FGH)
# Explanation:
## Step1: Find the angles of triangle ABC. As before, $\angle A = 30^\circ$, $\angle B = 60^\circ$, $\angle C = 90^\circ$ (since $180-(30 + 60)=90$).
## Step2: Find the angles of triangle FGH. In triangle FGH, $\angle F = 30^\circ$, $\angle G = 60^\circ$, so $\angle H=180 - 30 - 60 = 90^\circ$ (since $180-(30 + 60)=90$).
## Step3: Compare the angles. $\angle A=\angle F = 30^\circ$, $\angle B=\angle G = 60^\circ$ (and $\angle C=\angle H = 90^\circ$). So two pairs of corresponding angles are equal. By the angle - angle criterion, the triangles are similar.
# Answer:
Triangle ABC is similar to triangle FGH. In $\triangle ABC$, $\angle A = 30^\circ$, $\angle B = 60^\circ$, $\angle C = 90^\circ$. In $\triangle FGH$, $\angle F = 30^\circ$, $\angle G = 60^\circ$, $\angle H = 90^\circ$. Since $\angle A=\angle F$ and $\angle B=\angle G$ (two pairs of equal corresponding angles), by the angle - angle criterion, $\triangle ABC\sim\triangle FGH$.

### Part 4: Noticing about Angle Measures of Similar Triangles
# Explanation:
From the similar triangles (ABC and FGH), we saw that their corresponding angles are equal. So in similar triangles, the corresponding angles have equal measures.
# Answer:
In similar triangles (like $\triangle ABC$ and $\triangle FGH$), the corresponding angle measures are equal.

### Part 5: Filling in the Angle - Angle Criterion
# Explanation:
## Step1: Recall the angle - angle (AA) criterion for similar triangles. If two pairs of corresponding angles in a pair of triangles are congruent (equal in measure), then the two triangles are similar.
## Step2: Rewrite the criterion in own words. If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar (because the third angle will also be equal as the sum of angles in a triangle is $180^\circ$).
# Answer:
- If two pairs of corresponding angles in a pair of triangles are $\boldsymbol{\text{congruent (equal)}}$, then the two triangles are $\boldsymbol{\text{similar}}$.
- Rewrite: If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

### Part 6: Problem 1 (Using AA Criterion)
# Explanation:
## Step1: Find the third angle of the first triangle. In the first triangle, angles are $87^\circ$ and $58^\circ$. So the third angle $x=180-(87 + 58)=180 - 145 = 35^\circ$.
## Step2: Find the third angle of the second triangle. In the second triangle, angles are $87^\circ$ and $35^\circ$. So the third angle $y = 180-(87+35)=180 - 122 = 58^\circ$.
## Step3: Compare the angles. The first triangle has angles $87^\circ$, $58^\circ$, $35^\circ$ and the second triangle has angles $87^\circ$, $35^\circ$, $58^\circ$. So two pairs of angles are equal ($87^\circ = 87^\circ$ and $58^\circ=58^\circ$ or $87^\circ = 87^\circ$ and $35^\circ = 35^\circ$). By AA criterion, the triangles are similar.
# Answer:
The two triangles are similar. For the first triangle: $180-(87 + 58)=35^\circ$. For the second triangle: $180-(87 + 35)=58^\circ$. The angles of the first triangle are $87^\circ$, $58^\circ$, $35^\circ$ and the angles of the second triangle are $87^\circ$, $35^\circ$, $58^\circ$. Since two pairs of corresponding angles are equal, by the angle - angle criterion, the triangles are similar.

### Part 7: Problem 2 (Using AA Criterion)
# Explanation:
## Step1: Find the third angle of the first triangle. In the first triangle, angles are $80^\circ$ and $60^\circ$. So the third angle $x = 180-(80 + 60)=40^\circ$.
## Step2: Find the third angle of the second triangle. In the second triangle, angles are $80^\circ$ and $54^\circ$. So the third angle $y=180-(80 + 54)=46^\circ$.
## Step3: Compare the angles. The angles of the first triangle are $80^\circ$, $60^\circ$, $40^\circ$ and the angles of the second triangle are $80^\circ$, $54^\circ$, $46^\circ$. No two pairs of angles are equal. So the triangles are not similar.
# Answer:
The two triangles are not similar. For the first triangle: $180-(80 + 60)=40^\circ$. For the second triangle: $180-(80 + 54)=46^\circ$. The angle measures are $80^\circ$, $60^\circ$, $40^\circ$ (first) and $80^\circ$, $54^\circ$, $46^\circ$ (second). There are not two pairs of equal corresponding angles, so by the angle - angle criterion, they are not similar.

### Part 8: Problem 3 (Jace's and Ruthie's Triangles)
# Explanation:
## Step1: Find the angles of $\triangle ABC$. In $\triangle ABC$, $m\angle A = 53^\circ$, $m\angle B = 112^\circ$, so $m\angle C=180-(53 + 112)=180 - 165 = 15^\circ$.
## Step2: Find the angles of $\triangle RST$. In $\triangle RST$, $m\angle S = 112^\circ$, $m\angle T = 15^\circ$, so $m\angle R=180-(112 + 15)=180 - 127 = 53^\circ$.
## Step3: Compare the angles. In $\triangle ABC$, angles are $53^\circ$, $112^\circ$, $15^\circ$. In $\triangle RST$, angles are $53^\circ$, $112^\circ$, $15^\circ$. So $\angle A=\angle R = 53^\circ$, $\angle B=\angle S = 112^\circ$ (and $\angle C=\angle T = 15^\circ$). Two pairs of corresponding angles are equal. By AA criterion, the triangles are similar.
# Answer:
The two triangles are similar. In $\triangle ABC$: $m\angle C=180 - 53 - 112 = 15^\circ$. In $\triangle RST$: $m\angle R=180 - 112 - 15 = 53^\circ$. Since $m\angle A=m\angle R = 53^\circ$ and $m\angle B=m\angle S = 112^\circ$ (two pairs of equal corresponding angles), by the angle - angle criterion, $\triangle ABC\sim\triangle RST$.

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QUESTION IMAGE

Question

Explanation:

Part 1: Filling in the Blanks about Similar Figures

Step1: Recall the definition of similar figures. Similar figures have the same shape (they look alike in terms of their angles and the proportions of their sides) but can be different sizes. So the first blank is "shape".

Step2: Similar figures have proportional sides. This means that the ratios of corresponding sides are equal. So the second blank is "proportional" and the third blank is "equal".

Step1: Find the angles of triangle ABC. In triangle ABC, \(\angle A = 30^\circ\), \(\angle B = 60^\circ\), so \(\angle C=180 - 30 - 60=90^\circ\) (since the sum of angles in a triangle is \(180^\circ\)).

Step2: Find the angles of triangle DEF. In triangle DEF, \(\angle D = 35^\circ\), \(\angle E = 65^\circ\), so \(\angle F = 180 - 35 - 65 = 80^\circ\).

Step3: Compare the angles. The angles of ABC are \(30^\circ\), \(60^\circ\), \(90^\circ\) and angles of DEF are \(35^\circ\), \(65^\circ\), \(80^\circ\). No two pairs of angles are equal. So triangle ABC is not similar to triangle DEF.

Step1: Find the angles of triangle ABC. As before, \(\angle A = 30^\circ\), \(\angle B = 60^\circ\), \(\angle C = 90^\circ\) (since \(180-(30 + 60)=90\)).

Step2: Find the angles of triangle FGH. In triangle FGH, \(\angle F = 30^\circ\), \(\angle G = 60^\circ\), so \(\angle H=180 - 30 - 60 = 90^\circ\) (since \(180-(30 + 60)=90\)).

Step3: Compare the angles. \(\angle A=\angle F = 30^\circ\), \(\angle B=\angle G = 60^\circ\) (and \(\angle C=\angle H = 90^\circ\)). So two pairs of corresponding angles are equal. By the angle - angle criterion, the triangles are similar.

Answer:

Part 2: Determining Similarity of Triangles (ABC vs DEF)