Math / Geometry

QuestionGiven: $\overline{A B} \cong \overline{B C}, \angle A \cong \angle C$ and $\overline{B D}$ bisects $\overline{A C}$ . Prove: $\triangle A B D \cong \triangle C B D$ .

Studdy Solution

STEP 1

What is this asking? We're trying to prove that two triangles are congruent, given some information about their sides and angles. Watch out! Don't assume things just because they look equal in the diagram!
We need to use the given information and proper congruence rules.

STEP 2

1. State the givens
2. Prove the triangles are congruent

STEP 3

Alright, let's break down what we know!
We're given that $\overline{AB} \cong \overline{BC}$ .
This means the length of side AB is equal to the length of side BC.
That's a great starting point!

STEP 4

We also know that $\angle A \cong \angle C$ .
This tells us that angle A has the same measure as angle C.
Awesome!

STEP 5

Finally, we're given that $\overline{BD}$ bisects $\overline{AC}$ .
Remember, bisect means to cut into two equal parts.
So, $\overline{AD} \cong \overline{CD}$ .
This piece of information is crucial!

STEP 6

Now, let's look at what we've gathered.
We have $\overline{AB} \cong \overline{BC}$ , $\angle A \cong \angle C$ , and $\overline{AD} \cong \overline{CD}$ .
Do you see it?
These matching parts give us a congruence rule!

STEP 7

We have two pairs of congruent sides and a pair of congruent angles.
Specifically, we have two sides and the angle opposite one of the sides.
This is the Side-Angle-Side (SAS) congruence postulate!

STEP 8

Therefore, by SAS, we can confidently say that $\triangle ABD \cong \triangle CBD$ !
Boom!

STEP 9

We have successfully proven that $\triangle ABD \cong \triangle CBD$ using the SAS congruence postulate.

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QuestionGiven: $\overline{A B} \cong \overline{B C}, \angle A \cong \angle C$ and $\overline{B D}$ bisects $\overline{A C}$ . Prove: $\triangle A B D \cong \triangle C B D$ .

Studdy Solution

STEP 1

What is this asking? We're trying to prove that two triangles are congruent, given some information about their sides and angles. Watch out! Don't assume things just because they look equal in the diagram!
We need to use the given information and proper congruence rules.

STEP 2

1. State the givens
2. Prove the triangles are congruent

STEP 3

Alright, let's break down what we know!
We're given that $\overline{AB} \cong \overline{BC}$ .
This means the length of side AB is equal to the length of side BC.
That's a great starting point!

STEP 4

We also know that $\angle A \cong \angle C$ .
This tells us that angle A has the same measure as angle C.
Awesome!

STEP 5

Finally, we're given that $\overline{BD}$ bisects $\overline{AC}$ .
Remember, bisect means to cut into two equal parts.
So, $\overline{AD} \cong \overline{CD}$ .
This piece of information is crucial!

STEP 6

Now, let's look at what we've gathered.
We have $\overline{AB} \cong \overline{BC}$ , $\angle A \cong \angle C$ , and $\overline{AD} \cong \overline{CD}$ .
Do you see it?
These matching parts give us a congruence rule!

STEP 7

We have two pairs of congruent sides and a pair of congruent angles.
Specifically, we have two sides and the angle opposite one of the sides.
This is the Side-Angle-Side (SAS) congruence postulate!

STEP 8

Therefore, by SAS, we can confidently say that $\triangle ABD \cong \triangle CBD$ !
Boom!

STEP 9

We have successfully proven that $\triangle ABD \cong \triangle CBD$ using the SAS congruence postulate.

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QuestionGiven: AB‾≅BC‾,∠A≅∠C\overline{A B} \cong \overline{B C}, \angle A \cong \angle CAB≅BC,∠A≅∠C and BD‾\overline{B D}BD bisects AC‾\overline{A C}AC. Prove: △ABD≅△CBD\triangle A B D \cong \triangle C B D△ABD≅△CBD.

Studdy Solution

STEP 1

What is this asking? We're trying to prove that two triangles are congruent, given some information about their sides and angles. Watch out! Don't assume things just because they look equal in the diagram!We need to use the given information and proper congruence rules.

STEP 2

1. State the givens2. Prove the triangles are congruent

STEP 3

Alright, let's **break down** what we know!We're given that AB‾≅BC‾\overline{AB} \cong \overline{BC}AB≅BC.This means the length of side AB is equal to the length of side BC.That's a great starting point!

STEP 4

We also know that ∠A≅∠C\angle A \cong \angle C∠A≅∠C.This tells us that angle A has the same measure as angle C.Awesome!

STEP 5

Finally, we're given that BD‾\overline{BD}BD bisects AC‾\overline{AC}AC.Remember, **bisect** means to cut into two equal parts.So, AD‾≅CD‾\overline{AD} \cong \overline{CD}AD≅CD.This piece of information is crucial!

STEP 6

Now, let's look at what we've gathered.We have AB‾≅BC‾\overline{AB} \cong \overline{BC}AB≅BC, ∠A≅∠C\angle A \cong \angle C∠A≅∠C, and AD‾≅CD‾\overline{AD} \cong \overline{CD}AD≅CD.Do you see it?These matching parts give us a **congruence rule**!

STEP 7

We have two pairs of congruent sides and a pair of congruent angles.Specifically, we have two sides and the angle *opposite* one of the sides.This is the **Side-Angle-Side (SAS)** congruence postulate!

STEP 8

Therefore, by SAS, we can confidently say that △ABD≅△CBD\triangle ABD \cong \triangle CBD△ABD≅△CBD!Boom!

STEP 9

We have successfully proven that △ABD≅△CBD\triangle ABD \cong \triangle CBD△ABD≅△CBD using the SAS congruence postulate.

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Studdy solves anything!

Start learning now

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

QuestionGiven: $\overline{A B} \cong \overline{B C}, \angle A \cong \angle C$ and $\overline{B D}$ bisects $\overline{A C}$ . Prove: $\triangle A B D \cong \triangle C B D$ .

What is this asking? We're trying to prove that two triangles are congruent, given some information about their sides and angles. Watch out! Don't assume things just because they look equal in the diagram!
We need to use the given information and proper congruence rules.

1. State the givens
2. Prove the triangles are congruent

Alright, let's break down what we know!
We're given that $\overline{AB} \cong \overline{BC}$ .
This means the length of side AB is equal to the length of side BC.
That's a great starting point!

We also know that $\angle A \cong \angle C$ .
This tells us that angle A has the same measure as angle C.
Awesome!

Finally, we're given that $\overline{BD}$ bisects $\overline{AC}$ .
Remember, bisect means to cut into two equal parts.
So, $\overline{AD} \cong \overline{CD}$ .
This piece of information is crucial!

Now, let's look at what we've gathered.
We have $\overline{AB} \cong \overline{BC}$ , $\angle A \cong \angle C$ , and $\overline{AD} \cong \overline{CD}$ .
Do you see it?
These matching parts give us a congruence rule!

We have two pairs of congruent sides and a pair of congruent angles.
Specifically, we have two sides and the angle opposite one of the sides.
This is the Side-Angle-Side (SAS) congruence postulate!

Therefore, by SAS, we can confidently say that $\triangle ABD \cong \triangle CBD$ !
Boom!

We have successfully proven that $\triangle ABD \cong \triangle CBD$ using the SAS congruence postulate.