Math / Geometry

QuestionGiven $\angle E B C \cong \angle E C B, \overline{A E} \cong \overline{D E}$ Prove $\overline{A B} \cong \overline{D C}$
Statements
1. $\angle E B C=\angle E C B$
2. $A E=D E$
3. $E B=E C$
4. $\angle A E B=\angle D E C$
5. ABE : $\triangle D C E$
6. $A B=D C$
Reasons
1. $\square$ Click to add text
2. $\square$ Click to add text Click to add text $\square$ 3. $\square$ 4. 5.
Click to add text $\square$
6. $\square$

Studdy Solution

STEP 1

What is this asking? We're given that two angles and two sides are congruent, and we need to prove that two other sides of two triangles are also congruent! Watch out! Make sure to choose the correct triangle congruence theorem.
Don't mix up the sides and angles!

STEP 2

1. Fill in the Given Information
2. Determine Congruent Sides
3. Determine Congruent Angles
4. Prove Triangle Congruence
5. Conclude Side Congruence

STEP 3

We are given that $\angle EBC \cong \angle ECB$ .
This means their measures are equal: $\angle EBC = \angle ECB$ .
This is our first given piece of information.

STEP 4

We are also given that $\overline{AE} \cong \overline{DE}$ .
This means their lengths are equal: $AE = DE$ .
This is our second given piece of information.

STEP 5

Since $\angle EBC \cong \angle ECB$ , we know that $\triangle EBC$ is an isosceles triangle.

STEP 6

Because $\triangle EBC$ is isosceles, the sides opposite the congruent angles must also be congruent.
Therefore, $\overline{EB} \cong \overline{EC}$ , meaning $EB = EC$ .

STEP 7

Notice that $\angle AEB$ and $\angle DEC$ are vertical angles.

STEP 8

Since they are vertical angles, they are congruent: $\angle AEB \cong \angle DEC$ .
This means $\angle AEB = \angle DEC$ .

STEP 9

We have now shown that two sides and the included angle of $\triangle ABE$ are congruent to two sides and the included angle of $\triangle DCE$ .

STEP 10

Specifically, we have $AE = DE$ , $EB = EC$ , and $\angle AEB = \angle DEC$ .

STEP 11

This satisfies the Side-Angle-Side (SAS) congruence postulate.
Therefore, $\triangle ABE \cong \triangle DCE$ .

STEP 12

Since $\triangle ABE \cong \triangle DCE$ , all corresponding parts of these triangles are congruent.

STEP 13

This means $\overline{AB} \cong \overline{DC}$ , so $AB = DC$ , which is what we wanted to prove!

STEP 14

We have proven that $\overline{AB} \cong \overline{DC}$ .

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Studdy Solution

STEP 1

What is this asking? We're given that two angles and two sides are congruent, and we need to prove that two other sides of two triangles are also congruent! Watch out! Make sure to choose the correct triangle congruence theorem.
Don't mix up the sides and angles!

STEP 2

1. Fill in the Given Information
2. Determine Congruent Sides
3. Determine Congruent Angles
4. Prove Triangle Congruence
5. Conclude Side Congruence

STEP 3

We are given that $\angle EBC \cong \angle ECB$ .
This means their measures are equal: $\angle EBC = \angle ECB$ .
This is our first given piece of information.

STEP 4

We are also given that $\overline{AE} \cong \overline{DE}$ .
This means their lengths are equal: $AE = DE$ .
This is our second given piece of information.

STEP 5

Since $\angle EBC \cong \angle ECB$ , we know that $\triangle EBC$ is an isosceles triangle.

STEP 6

Because $\triangle EBC$ is isosceles, the sides opposite the congruent angles must also be congruent.
Therefore, $\overline{EB} \cong \overline{EC}$ , meaning $EB = EC$ .

STEP 7

Notice that $\angle AEB$ and $\angle DEC$ are vertical angles.

STEP 8

Since they are vertical angles, they are congruent: $\angle AEB \cong \angle DEC$ .
This means $\angle AEB = \angle DEC$ .

STEP 9

We have now shown that two sides and the included angle of $\triangle ABE$ are congruent to two sides and the included angle of $\triangle DCE$ .

STEP 10

Specifically, we have $AE = DE$ , $EB = EC$ , and $\angle AEB = \angle DEC$ .

STEP 11

This satisfies the Side-Angle-Side (SAS) congruence postulate.
Therefore, $\triangle ABE \cong \triangle DCE$ .

STEP 12

Since $\triangle ABE \cong \triangle DCE$ , all corresponding parts of these triangles are congruent.

STEP 13

This means $\overline{AB} \cong \overline{DC}$ , so $AB = DC$ , which is what we wanted to prove!

STEP 14

We have proven that $\overline{AB} \cong \overline{DC}$ .

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Studdy Solution

STEP 1

What is this asking? We're given that two angles and two sides are congruent, and we need to prove that two other sides of two triangles are also congruent! Watch out! Make sure to choose the correct triangle congruence theorem.Don't mix up the sides and angles!

STEP 2

1. Fill in the Given Information2. Determine Congruent Sides3. Determine Congruent Angles4. Prove Triangle Congruence5. Conclude Side Congruence

STEP 3

We are given that ∠EBC≅∠ECB\angle EBC \cong \angle ECB∠EBC≅∠ECB.This means their measures are equal: ∠EBC=∠ECB\angle EBC = \angle ECB∠EBC=∠ECB.This is our **first given** piece of information.

STEP 4

We are also given that AE‾≅DE‾\overline{AE} \cong \overline{DE}AE≅DE.This means their lengths are equal: AE=DEAE = DEAE=DE.This is our **second given** piece of information.

STEP 5

Since ∠EBC≅∠ECB\angle EBC \cong \angle ECB∠EBC≅∠ECB, we know that △EBC\triangle EBC△EBC is an **isosceles triangle**.

STEP 6

Because △EBC\triangle EBC△EBC is isosceles, the sides opposite the congruent angles must also be congruent.Therefore, EB‾≅EC‾\overline{EB} \cong \overline{EC}EB≅EC, meaning EB=ECEB = ECEB=EC.

STEP 7

Notice that ∠AEB\angle AEB∠AEB and ∠DEC\angle DEC∠DEC are **vertical angles**.

STEP 8

Since they are vertical angles, they are congruent: ∠AEB≅∠DEC\angle AEB \cong \angle DEC∠AEB≅∠DEC.This means ∠AEB=∠DEC\angle AEB = \angle DEC∠AEB=∠DEC.

STEP 9

We have now shown that two sides and the included angle of △ABE\triangle ABE△ABE are congruent to two sides and the included angle of △DCE\triangle DCE△DCE.

STEP 10

Specifically, we have AE=DEAE = DEAE=DE, EB=ECEB = ECEB=EC, and ∠AEB=∠DEC\angle AEB = \angle DEC∠AEB=∠DEC.

STEP 11

This satisfies the **Side-Angle-Side (SAS)** congruence postulate.Therefore, △ABE≅△DCE\triangle ABE \cong \triangle DCE△ABE≅△DCE.

STEP 12

Since △ABE≅△DCE\triangle ABE \cong \triangle DCE△ABE≅△DCE, all corresponding parts of these triangles are congruent.

STEP 13

This means AB‾≅DC‾\overline{AB} \cong \overline{DC}AB≅DC, so AB=DCAB = DCAB=DC, which is what we wanted to prove!

STEP 14

We have proven that AB‾≅DC‾\overline{AB} \cong \overline{DC}AB≅DC.

Get Studdy

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.

What is this asking? We're given that two angles and two sides are congruent, and we need to prove that two other sides of two triangles are also congruent! Watch out! Make sure to choose the correct triangle congruence theorem.
Don't mix up the sides and angles!

1. Fill in the Given Information
2. Determine Congruent Sides
3. Determine Congruent Angles
4. Prove Triangle Congruence
5. Conclude Side Congruence

We are given that $\angle EBC \cong \angle ECB$ .
This means their measures are equal: $\angle EBC = \angle ECB$ .
This is our first given piece of information.

We are also given that $\overline{AE} \cong \overline{DE}$ .
This means their lengths are equal: $AE = DE$ .
This is our second given piece of information.

Since $\angle EBC \cong \angle ECB$ , we know that $\triangle EBC$ is an isosceles triangle.

Because $\triangle EBC$ is isosceles, the sides opposite the congruent angles must also be congruent.
Therefore, $\overline{EB} \cong \overline{EC}$ , meaning $EB = EC$ .

Notice that $\angle AEB$ and $\angle DEC$ are vertical angles.

Since they are vertical angles, they are congruent: $\angle AEB \cong \angle DEC$ .
This means $\angle AEB = \angle DEC$ .

We have now shown that two sides and the included angle of $\triangle ABE$ are congruent to two sides and the included angle of $\triangle DCE$ .

Specifically, we have $AE = DE$ , $EB = EC$ , and $\angle AEB = \angle DEC$ .

This satisfies the Side-Angle-Side (SAS) congruence postulate.
Therefore, $\triangle ABE \cong \triangle DCE$ .

Since $\triangle ABE \cong \triangle DCE$ , all corresponding parts of these triangles are congruent.

This means $\overline{AB} \cong \overline{DC}$ , so $AB = DC$ , which is what we wanted to prove!

We have proven that $\overline{AB} \cong \overline{DC}$ .