Triangle Similarity Theorems and Pythagorean Theorem - Geometry
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What property of dilations is used in proving AA similarity?
What property of dilations is used in proving AA similarity?
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Dilations preserve angle measure and scale all lengths by $k$. Angles stay the same while sides scale uniformly.
Dilations preserve angle measure and scale all lengths by $k$. Angles stay the same while sides scale uniformly.
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Choose the correct correspondence: $\triangle ABC \sim \triangle RST$. Which angle matches $\angle B$?
Choose the correct correspondence: $\triangle ABC \sim \triangle RST$. Which angle matches $\angle B$?
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$\angle S$. Second angle corresponds to second angle in the order.
$\angle S$. Second angle corresponds to second angle in the order.
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If $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{2}{5}$, what is $k$ from $ABC$ to $DEF$?
If $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{2}{5}$, what is $k$ from $ABC$ to $DEF$?
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$k=\frac{5}{2}$. Scale factor is the reciprocal: $k = \frac{1}{\frac{2}{5}} = \frac{5}{2}$.
$k=\frac{5}{2}$. Scale factor is the reciprocal: $k = \frac{1}{\frac{2}{5}} = \frac{5}{2}$.
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If $\triangle ABC \sim \triangle DEF$ and $AC=15$, $DF=20$, what is $k$ from $ABC$ to $DEF$?
If $\triangle ABC \sim \triangle DEF$ and $AC=15$, $DF=20$, what is $k$ from $ABC$ to $DEF$?
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$k=\frac{20}{15}=\frac{4}{3}$. Scale factor is the ratio of corresponding sides.
$k=\frac{20}{15}=\frac{4}{3}$. Scale factor is the ratio of corresponding sides.
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If $\triangle ABC \sim \triangle DEF$ and $k=\frac{1}{3}$ from $ABC$ to $DEF$, what is $EF$ when $BC=12$?
If $\triangle ABC \sim \triangle DEF$ and $k=\frac{1}{3}$ from $ABC$ to $DEF$, what is $EF$ when $BC=12$?
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$EF=4$. Multiply corresponding side by scale factor: $12 \times \frac{1}{3} = 4$.
$EF=4$. Multiply corresponding side by scale factor: $12 \times \frac{1}{3} = 4$.
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If $\triangle ABC \sim \triangle DEF$ and $k=2$, what is $DE$ when $AB=7$?
If $\triangle ABC \sim \triangle DEF$ and $k=2$, what is $DE$ when $AB=7$?
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$DE=14$. Multiply corresponding side by scale factor: $7 \times 2 = 14$.
$DE=14$. Multiply corresponding side by scale factor: $7 \times 2 = 14$.
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If $\triangle ABC \sim \triangle DEF$ and $AB=8$, $DE=12$, what is the scale factor from $ABC$ to $DEF$?
If $\triangle ABC \sim \triangle DEF$ and $AB=8$, $DE=12$, what is the scale factor from $ABC$ to $DEF$?
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$k=\frac{12}{8}=\frac{3}{2}$. Scale factor is the ratio of corresponding sides.
$k=\frac{12}{8}=\frac{3}{2}$. Scale factor is the ratio of corresponding sides.
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If $\angle A \cong \angle D$ and $\angle B \cong \angle E$, which angles must also be congruent?
If $\angle A \cong \angle D$ and $\angle B \cong \angle E$, which angles must also be congruent?
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$\angle C \cong \angle F$. The third pair must be congruent by the angle sum theorem.
$\angle C \cong \angle F$. The third pair must be congruent by the angle sum theorem.
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If two angles of one triangle match two angles of another, what can you conclude about the third angles?
If two angles of one triangle match two angles of another, what can you conclude about the third angles?
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The third angles are congruent. The angle sum theorem ensures the third angles are equal.
The third angles are congruent. The angle sum theorem ensures the third angles are equal.
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Find and correct the statement: "Rigid motions change distances by a scale factor."
Find and correct the statement: "Rigid motions change distances by a scale factor."
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Correct: rigid motions preserve distances exactly. Rigid motions preserve all distances exactly.
Correct: rigid motions preserve distances exactly. Rigid motions preserve all distances exactly.
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If $\angle A \cong \angle D$ and $\angle B \cong \angle E$, what similarity statement matches the order?
If $\angle A \cong \angle D$ and $\angle B \cong \angle E$, what similarity statement matches the order?
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$\triangle ABC \sim \triangle DEF$. The order of vertices shows the correspondence.
$\triangle ABC \sim \triangle DEF$. The order of vertices shows the correspondence.
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Which option proves similarity: $\angle A \cong \angle D$ and $\angle C \cong \angle F$?
Which option proves similarity: $\angle A \cong \angle D$ and $\angle C \cong \angle F$?
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AA similarity applies, so the triangles are similar. Two pairs of congruent angles prove similarity by AA.
AA similarity applies, so the triangles are similar. Two pairs of congruent angles prove similarity by AA.
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Find the missing ratio if $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{3}{4}$: what is $\frac{BC}{EF}$?
Find the missing ratio if $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{3}{4}$: what is $\frac{BC}{EF}$?
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$\frac{3}{4}$. All corresponding ratios are equal in similar triangles.
$\frac{3}{4}$. All corresponding ratios are equal in similar triangles.
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Choose the correct correspondence: $\triangle ABC \sim \triangle RST$. Which side matches $\overline{BC}$?
Choose the correct correspondence: $\triangle ABC \sim \triangle RST$. Which side matches $\overline{BC}$?
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$\overline{ST}$. Second and third sides correspond by position.
$\overline{ST}$. Second and third sides correspond by position.
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Which option proves similarity: $\angle A \cong \angle D$ and $\angle C \cong \angle F$?
Which option proves similarity: $\angle A \cong \angle D$ and $\angle C \cong \angle F$?
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AA similarity applies, so the triangles are similar. Two pairs of congruent angles prove similarity by AA.
AA similarity applies, so the triangles are similar. Two pairs of congruent angles prove similarity by AA.
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Identify the conclusion: $\angle A \cong \angle D$ and $\angle B \cong \angle E$ for two triangles.
Identify the conclusion: $\angle A \cong \angle D$ and $\angle B \cong \angle E$ for two triangles.
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The triangles are similar by AA. Two pairs of congruent angles satisfy the AA criterion.
The triangles are similar by AA. Two pairs of congruent angles satisfy the AA criterion.
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If $m\angle D=50^\circ$ and $m\angle E=60^\circ$, what is $m\angle F$?
If $m\angle D=50^\circ$ and $m\angle E=60^\circ$, what is $m\angle F$?
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$70^\circ$. Angles sum to $180^\circ$: $180^\circ - 50^\circ - 60^\circ = 70^\circ$.
$70^\circ$. Angles sum to $180^\circ$: $180^\circ - 50^\circ - 60^\circ = 70^\circ$.
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What is the third angle if a triangle has angles $28^\circ$ and $92^\circ$?
What is the third angle if a triangle has angles $28^\circ$ and $92^\circ$?
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$60^\circ$. Third angle: $180^\circ - 28^\circ - 92^\circ = 60^\circ$.
$60^\circ$. Third angle: $180^\circ - 28^\circ - 92^\circ = 60^\circ$.
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If $m\angle A=50^\circ$, $m\angle B=60^\circ$, what is $m\angle C$?
If $m\angle A=50^\circ$, $m\angle B=60^\circ$, what is $m\angle C$?
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$70^\circ$. Angles sum to $180^\circ$: $180^\circ - 50^\circ - 60^\circ = 70^\circ$.
$70^\circ$. Angles sum to $180^\circ$: $180^\circ - 50^\circ - 60^\circ = 70^\circ$.
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What is the scale factor from $\triangle DEF$ to $\triangle ABC$ if $DE=10$ corresponds to $AB=4$?
What is the scale factor from $\triangle DEF$ to $\triangle ABC$ if $DE=10$ corresponds to $AB=4$?
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$k=\frac{4}{10}=\frac{2}{5}$. Scale factor is the ratio from first to second triangle.
$k=\frac{4}{10}=\frac{2}{5}$. Scale factor is the ratio from first to second triangle.
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What is the scale factor from $\triangle ABC$ to $\triangle DEF$ if $AB=6$ and $DE=9$ correspond?
What is the scale factor from $\triangle ABC$ to $\triangle DEF$ if $AB=6$ and $DE=9$ correspond?
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$k=\frac{9}{6}=\frac{3}{2}$. Scale factor is the ratio of corresponding lengths.
$k=\frac{9}{6}=\frac{3}{2}$. Scale factor is the ratio of corresponding lengths.
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Identify the missing step: rigid motion maps one vertex and one side; what is done next to match size?
Identify the missing step: rigid motion maps one vertex and one side; what is done next to match size?
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Apply a dilation with scale factor $k$. Dilation adjusts size after rigid motions align position.
Apply a dilation with scale factor $k$. Dilation adjusts size after rigid motions align position.
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What is the key conclusion of AA similarity about angle measures?
What is the key conclusion of AA similarity about angle measures?
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All corresponding angles are congruent. Similar triangles have all corresponding angles congruent.
All corresponding angles are congruent. Similar triangles have all corresponding angles congruent.
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What is the key conclusion of AA similarity about side lengths?
What is the key conclusion of AA similarity about side lengths?
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Corresponding sides are proportional. Similar triangles have proportional corresponding sides.
Corresponding sides are proportional. Similar triangles have proportional corresponding sides.
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What does it mean for two angles to be congruent?
What does it mean for two angles to be congruent?
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They have equal measure. Congruent angles have identical measures.
They have equal measure. Congruent angles have identical measures.
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What is the minimum angle information needed to prove two triangles similar by AA?
What is the minimum angle information needed to prove two triangles similar by AA?
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Two pairs of corresponding congruent angles. AA requires exactly two pairs of congruent corresponding angles.
Two pairs of corresponding congruent angles. AA requires exactly two pairs of congruent corresponding angles.
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What is the correct proportionality statement if $\triangle ABC \sim \triangle DEF$?
What is the correct proportionality statement if $\triangle ABC \sim \triangle DEF$?
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$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$. This shows the correct order of corresponding sides.
$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$. This shows the correct order of corresponding sides.
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In $\triangle ABC \sim \triangle DEF$, which side corresponds to $\overline{AC}$?
In $\triangle ABC \sim \triangle DEF$, which side corresponds to $\overline{AC}$?
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$\overline{DF}$. Corresponding sides match by position in the similarity statement.
$\overline{DF}$. Corresponding sides match by position in the similarity statement.
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In $\triangle ABC \sim \triangle DEF$, which angle corresponds to $\angle B$?
In $\triangle ABC \sim \triangle DEF$, which angle corresponds to $\angle B$?
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$\angle E$. Corresponding angles match by position in the similarity statement.
$\angle E$. Corresponding angles match by position in the similarity statement.
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What is the relationship between corresponding side ratios in similar triangles?
What is the relationship between corresponding side ratios in similar triangles?
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All corresponding side ratios are equal to the same constant $k$. The constant ratio $k$ is the scale factor.
All corresponding side ratios are equal to the same constant $k$. The constant ratio $k$ is the scale factor.
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