Math / Geometry

QuestionIn triangles $P Q R$ and $U V W$ , angles $Q$ and $V$ each have measure $75^{\circ}, P Q=9$ , and $U V=27$ . Which additional piece of information is sufficient to prove that triangle $P Q R$ is similar to triangle $U V W$ ?

Studdy Solution

STEP 1

What is this asking? We're checking if two triangles are similar, knowing one angle and one side in each triangle, and we need to figure out what other info we need to prove they're similar. Watch out! Remember, similar triangles have the same angles and proportional sides, not necessarily equal sides!

STEP 2

1. Similarity Conditions
2. Analyze Options

STEP 3

Alright, awesome students!
Let's dive into the world of similar triangles!
We already know that angle $Q$ and angle $V$ are equal, both measuring $75^\circ$ .
That's a fantastic start!

STEP 4

Now, remember the magical ways to prove triangle similarity: Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS).
Since we only have one angle confirmed as equal, SSS is out the window.
We need either another angle or the right side ratios for SAS.

STEP 5

Let's look at what we do know.
We have $PQ = 9$ and $UV = 27$ .
Notice that $UV = 3 \cdot PQ$ .
So, if these triangles are similar, the ratio of corresponding sides will be $1:3$ or $3:1$ .

STEP 6

For SAS, we'd need the sides around the $75^\circ$ angles to be in that $1:3$ ratio.
So, we'd need $PR:UW = 1:3$ (or $3:1$ ) and $QR:VW = 1:3$ (or $3:1$ ).

STEP 7

For AA, we'd need another pair of matching angles.
If we knew that $\angle P = \angle U$ or $\angle R = \angle W$ , we'd be golden!

STEP 8

Knowing the ratio of $PR$ to $UW$ or $QR$ to $VW$ is $1:3$ would be enough for SAS similarity.
Alternatively, knowing $\angle P = \angle U$ or $\angle R = \angle W$ would give us AA similarity.

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QuestionIn triangles $P Q R$ and $U V W$ , angles $Q$ and $V$ each have measure $75^{\circ}, P Q=9$ , and $U V=27$ . Which additional piece of information is sufficient to prove that triangle $P Q R$ is similar to triangle $U V W$ ?

Studdy Solution

STEP 1

STEP 2

1. Similarity Conditions
2. Analyze Options

STEP 3

Alright, awesome students!
Let's dive into the world of similar triangles!
We already know that angle $Q$ and angle $V$ are equal, both measuring $75^\circ$ .
That's a fantastic start!

STEP 4

Now, remember the magical ways to prove triangle similarity: Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS).
Since we only have one angle confirmed as equal, SSS is out the window.
We need either another angle or the right side ratios for SAS.

STEP 5

Let's look at what we do know.
We have $PQ = 9$ and $UV = 27$ .
Notice that $UV = 3 \cdot PQ$ .
So, if these triangles are similar, the ratio of corresponding sides will be $1:3$ or $3:1$ .

STEP 6

For SAS, we'd need the sides around the $75^\circ$ angles to be in that $1:3$ ratio.
So, we'd need $PR:UW = 1:3$ (or $3:1$ ) and $QR:VW = 1:3$ (or $3:1$ ).

STEP 7

For AA, we'd need another pair of matching angles.
If we knew that $\angle P = \angle U$ or $\angle R = \angle W$ , we'd be golden!

STEP 8

Knowing the ratio of $PR$ to $UW$ or $QR$ to $VW$ is $1:3$ would be enough for SAS similarity.
Alternatively, knowing $\angle P = \angle U$ or $\angle R = \angle W$ would give us AA similarity.

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QuestionIn triangles PQRP Q RPQR and UVWU V WUVW, angles QQQ and VVV each have measure 75∘,PQ=975^{\circ}, P Q=975∘,PQ=9, and UV=27U V=27UV=27. Which additional piece of information is sufficient to prove that triangle PQRP Q RPQR is similar to triangle UVWU V WUVW ?

Studdy Solution

STEP 1

STEP 2

1. Similarity Conditions2. Analyze Options

STEP 3

Alright, awesome students!Let's dive into the world of similar triangles!We already know that angle QQQ and angle VVV are equal, both measuring 75∘75^\circ75∘.That's a fantastic start!

STEP 4

Now, remember the magical ways to prove triangle similarity: **Angle-Angle (AA)**, **Side-Angle-Side (SAS)**, and **Side-Side-Side (SSS)**.Since we only have one angle confirmed as equal, SSS is out the window.We need either another angle or the right side ratios for SAS.

STEP 5

Let's look at what we *do* know.We have PQ=9PQ = 9PQ=9 and UV=27UV = 27UV=27.Notice that UV=3⋅PQUV = 3 \cdot PQUV=3⋅PQ.So, if these triangles *are* similar, the ratio of corresponding sides will be 1:31:31:3 or 3:13:13:1.

STEP 6

For SAS, we'd need the sides around the 75∘75^\circ75∘ angles to be in that 1:31:31:3 ratio.So, we'd need PR:UW=1:3PR:UW = 1:3PR:UW=1:3 (or 3:13:13:1) *and* QR:VW=1:3QR:VW = 1:3QR:VW=1:3 (or 3:13:13:1).

STEP 7

For AA, we'd need another pair of matching angles.If we knew that ∠P=∠U\angle P = \angle U∠P=∠U or ∠R=∠W\angle R = \angle W∠R=∠W, we'd be golden!

STEP 8

Knowing the ratio of PRPRPR to UWUWUW *or* QRQRQR to VWVWVW is 1:31:31:3 would be enough for SAS similarity.Alternatively, knowing ∠P=∠U\angle P = \angle U∠P=∠U or ∠R=∠W\angle R = \angle W∠R=∠W would give us AA similarity.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionIn triangles $P Q R$ and $U V W$ , angles $Q$ and $V$ each have measure $75^{\circ}, P Q=9$ , and $U V=27$ . Which additional piece of information is sufficient to prove that triangle $P Q R$ is similar to triangle $U V W$ ?

1. Similarity Conditions
2. Analyze Options

Alright, awesome students!
Let's dive into the world of similar triangles!
We already know that angle $Q$ and angle $V$ are equal, both measuring $75^\circ$ .
That's a fantastic start!

Now, remember the magical ways to prove triangle similarity: Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS).
Since we only have one angle confirmed as equal, SSS is out the window.
We need either another angle or the right side ratios for SAS.

Let's look at what we do know.
We have $PQ = 9$ and $UV = 27$ .
Notice that $UV = 3 \cdot PQ$ .
So, if these triangles are similar, the ratio of corresponding sides will be $1:3$ or $3:1$ .

For SAS, we'd need the sides around the $75^\circ$ angles to be in that $1:3$ ratio.
So, we'd need $PR:UW = 1:3$ (or $3:1$ ) and $QR:VW = 1:3$ (or $3:1$ ).

For AA, we'd need another pair of matching angles.
If we knew that $\angle P = \angle U$ or $\angle R = \angle W$ , we'd be golden!

Knowing the ratio of $PR$ to $UW$ or $QR$ to $VW$ is $1:3$ would be enough for SAS similarity.
Alternatively, knowing $\angle P = \angle U$ or $\angle R = \angle W$ would give us AA similarity.