Math / Geometry

QuestionGiven: $\overline{A C} \cong \overline{B D}$ and $\angle C A B \cong \angle D B A$ . Prove: $\triangle A B C \cong \triangle B A D$ .
Step
1 try
Statement $\overline{A C} \cong \overline{B D}$ $\angle C A B \cong \angle D B A$
Reason
Given
Type of Statement
Note: $\overline{A C}$ and $\overline{D B}$ are segments.

Studdy Solution

STEP 1

What is this asking? We're given that two segments, AC and BD, are congruent, and two angles, CAB and DBA, are also congruent.
We need to prove that triangles ABC and BAD are congruent. Watch out! Make sure you're matching the correct sides and angles when applying congruence postulates!
Don't get tricked by the order of the letters in the triangle names.

STEP 2

1. Identify Congruent Parts
2. Find the Shared Side
3. Apply SAS Congruence

STEP 3

We're given that segment AC is congruent to segment BD, so let's write that down mathematically: $\overline{AC} \cong \overline{BD}$ .
This is a key piece of our puzzle!

STEP 4

We also know that angle CAB is congruent to angle DBA.
Let's express this as $\angle CAB \cong \angle DBA$ .
This is another important clue!

STEP 5

Notice that both triangles ABC and BAD share the side AB.
This means that segment AB is congruent to itself!
We can write this as $\overline{AB} \cong \overline{AB}$ using the reflexive property of congruence.
This shared side is the missing link we need!

STEP 6

Now, let's put it all together.
We have two pairs of congruent sides and a pair of congruent angles.
Specifically, we have $\overline{AC} \cong \overline{BD}$ , $\angle CAB \cong \angle DBA$ , and $\overline{AB} \cong \overline{AB}$ .

STEP 7

Look closely!
The congruent angle is between the two congruent sides in both triangles.
This is the Side-Angle-Side (SAS) congruence postulate!

STEP 8

Therefore, by SAS, we can confidently say that triangle ABC is congruent to triangle BAD.
We write this as $\triangle ABC \cong \triangle BAD$ .
Boom!

STEP 9

We have successfully proven that $\triangle ABC \cong \triangle BAD$ using the SAS congruence postulate.

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Studdy Solution

STEP 1

STEP 2

1. Identify Congruent Parts
2. Find the Shared Side
3. Apply SAS Congruence

STEP 3

We're given that segment AC is congruent to segment BD, so let's write that down mathematically: $\overline{AC} \cong \overline{BD}$ .
This is a key piece of our puzzle!

STEP 4

We also know that angle CAB is congruent to angle DBA.
Let's express this as $\angle CAB \cong \angle DBA$ .
This is another important clue!

STEP 5

Notice that both triangles ABC and BAD share the side AB.
This means that segment AB is congruent to itself!
We can write this as $\overline{AB} \cong \overline{AB}$ using the reflexive property of congruence.
This shared side is the missing link we need!

STEP 6

Now, let's put it all together.
We have two pairs of congruent sides and a pair of congruent angles.
Specifically, we have $\overline{AC} \cong \overline{BD}$ , $\angle CAB \cong \angle DBA$ , and $\overline{AB} \cong \overline{AB}$ .

STEP 7

Look closely!
The congruent angle is between the two congruent sides in both triangles.
This is the Side-Angle-Side (SAS) congruence postulate!

STEP 8

Therefore, by SAS, we can confidently say that triangle ABC is congruent to triangle BAD.
We write this as $\triangle ABC \cong \triangle BAD$ .
Boom!

STEP 9

We have successfully proven that $\triangle ABC \cong \triangle BAD$ using the SAS congruence postulate.

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Studdy Solution

STEP 1

STEP 2

1. Identify Congruent Parts2. Find the Shared Side3. Apply SAS Congruence

STEP 3

We're given that segment AC is congruent to segment BD, so let's write that down mathematically: AC‾≅BD‾\overline{AC} \cong \overline{BD}AC≅BD.This is a **key piece** of our puzzle!

STEP 4

We also know that angle CAB is congruent to angle DBA.Let's express this as ∠CAB≅∠DBA\angle CAB \cong \angle DBA∠CAB≅∠DBA.This is another **important clue**!

STEP 5

Notice that both triangles ABC and BAD share the side AB.This means that segment AB is congruent to itself!We can write this as AB‾≅AB‾\overline{AB} \cong \overline{AB}AB≅AB using the **reflexive property** of congruence.This **shared side** is the **missing link** we need!

STEP 6

Now, let's put it all together.We have two pairs of congruent sides and a pair of congruent angles.Specifically, we have AC‾≅BD‾\overline{AC} \cong \overline{BD}AC≅BD, ∠CAB≅∠DBA\angle CAB \cong \angle DBA∠CAB≅∠DBA, and AB‾≅AB‾\overline{AB} \cong \overline{AB}AB≅AB.

STEP 7

Look closely!The congruent angle is *between* the two congruent sides in both triangles.This is the **Side-Angle-Side (SAS)** congruence postulate!

STEP 8

Therefore, by SAS, we can confidently say that triangle ABC is congruent to triangle BAD.We write this as △ABC≅△BAD\triangle ABC \cong \triangle BAD△ABC≅△BAD.Boom!

STEP 9

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

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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.

1. Identify Congruent Parts
2. Find the Shared Side
3. Apply SAS Congruence

We're given that segment AC is congruent to segment BD, so let's write that down mathematically: $\overline{AC} \cong \overline{BD}$ .
This is a key piece of our puzzle!

We also know that angle CAB is congruent to angle DBA.
Let's express this as $\angle CAB \cong \angle DBA$ .
This is another important clue!

Notice that both triangles ABC and BAD share the side AB.
This means that segment AB is congruent to itself!
We can write this as $\overline{AB} \cong \overline{AB}$ using the reflexive property of congruence.
This shared side is the missing link we need!

Now, let's put it all together.
We have two pairs of congruent sides and a pair of congruent angles.
Specifically, we have $\overline{AC} \cong \overline{BD}$ , $\angle CAB \cong \angle DBA$ , and $\overline{AB} \cong \overline{AB}$ .

Look closely!
The congruent angle is between the two congruent sides in both triangles.
This is the Side-Angle-Side (SAS) congruence postulate!

Therefore, by SAS, we can confidently say that triangle ABC is congruent to triangle BAD.
We write this as $\triangle ABC \cong \triangle BAD$ .
Boom!

We have successfully proven that $\triangle ABC \cong \triangle BAD$ using the SAS congruence postulate.