Math / Geometry

Questionse the given information to prove that $\triangle G E D \cong \triangle G F D$ .
Prove: $\triangle G E D \cong \triangle G F D$ Send To Proof Statement Reason
1 $\square$ Reason? Save For Later

Studdy Solution

STEP 1

What is this asking? We need to show that two triangles, GED and GFD, are congruent, knowing they share a side and have congruent sides DE and DF. Watch out! Don't assume anything beyond what's given!
We only know those two sides are equal, and the shared side.

STEP 2

1. State the givens
2. Establish shared side
3. Use Hypotenuse-Leg Theorem

STEP 3

We are given that $DE \cong DF$ .
This means the lengths of sides $DE$ and $DF$ are equal.
This is a key fact we'll use later!

STEP 4

We're also told that triangles GED and GFD are right triangles.
This means they each have a right angle, specifically, angles GED and GFD are right angles.

STEP 5

The triangles share the side $GD$ .
Because it's the same segment in both triangles, we know $GD \cong GD$ .
This is called the reflexive property!
We now have two pairs of congruent sides between the triangles!

STEP 6

In right triangles GED and GFD, sides GE and GF are opposite the right angles.
This makes GE and GF the hypotenuses of their respective triangles.

STEP 7

We've shown that the triangles have congruent legs ( $DE \cong DF$ ) and a congruent hypotenuse ( $GD \cong GD$ ).
This is precisely what the Hypotenuse-Leg (HL) Theorem states!
If two right triangles have congruent hypotenuses and a pair of congruent legs, then the triangles are congruent.

STEP 8

Therefore, by the HL Theorem, we can confidently say $\triangle GED \cong \triangle GFD$ !

STEP 9

We have successfully proven that $\triangle GED \cong \triangle GFD$ using the Hypotenuse-Leg Theorem!

Was this helpful?

Questionse the given information to prove that $\triangle G E D \cong \triangle G F D$ .
Prove: $\triangle G E D \cong \triangle G F D$ Send To Proof Statement Reason
1 $\square$ Reason? Save For Later

Studdy Solution

STEP 1

What is this asking? We need to show that two triangles, GED and GFD, are congruent, knowing they share a side and have congruent sides DE and DF. Watch out! Don't assume anything beyond what's given!
We only know those two sides are equal, and the shared side.

STEP 2

1. State the givens
2. Establish shared side
3. Use Hypotenuse-Leg Theorem

STEP 3

We are given that $DE \cong DF$ .
This means the lengths of sides $DE$ and $DF$ are equal.
This is a key fact we'll use later!

STEP 4

We're also told that triangles GED and GFD are right triangles.
This means they each have a right angle, specifically, angles GED and GFD are right angles.

STEP 5

The triangles share the side $GD$ .
Because it's the same segment in both triangles, we know $GD \cong GD$ .
This is called the reflexive property!
We now have two pairs of congruent sides between the triangles!

STEP 6

In right triangles GED and GFD, sides GE and GF are opposite the right angles.
This makes GE and GF the hypotenuses of their respective triangles.

STEP 7

We've shown that the triangles have congruent legs ( $DE \cong DF$ ) and a congruent hypotenuse ( $GD \cong GD$ ).
This is precisely what the Hypotenuse-Leg (HL) Theorem states!
If two right triangles have congruent hypotenuses and a pair of congruent legs, then the triangles are congruent.

STEP 8

Therefore, by the HL Theorem, we can confidently say $\triangle GED \cong \triangle GFD$ !

STEP 9

We have successfully proven that $\triangle GED \cong \triangle GFD$ using the Hypotenuse-Leg Theorem!

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.

Questionse the given information to prove that △GED≅△GFD\triangle G E D \cong \triangle G F D△GED≅△GFD.Prove: △GED≅△GFD\triangle G E D \cong \triangle G F D△GED≅△GFD Send To Proof Statement Reason1 □\square□ Reason? Save For Later

Studdy Solution

STEP 1

What is this asking? We need to show that two triangles, GED and GFD, are congruent, knowing they share a side and have congruent sides DE and DF. Watch out! Don't assume anything beyond what's given!We only know *those* two sides are equal, and the shared side.

STEP 2

1. State the givens2. Establish shared side3. Use Hypotenuse-Leg Theorem

STEP 3

We are given that DE≅DFDE \cong DFDE≅DF.This means the lengths of sides DEDEDE and DFDFDF are **equal**.This is a **key fact** we'll use later!

STEP 4

We're also told that triangles GED and GFD are **right triangles**.This means they each have a **right angle**, specifically, angles GED and GFD are right angles.

STEP 5

The triangles share the side GDGDGD.Because it's the same segment in both triangles, we know GD≅GDGD \cong GDGD≅GD.This is called the **reflexive property**!We now have **two pairs of congruent sides** between the triangles!

STEP 6

In right triangles GED and GFD, sides GE and GF are opposite the right angles.This makes GE and GF the **hypotenuses** of their respective triangles.

STEP 7

STEP 8

Therefore, by the HL Theorem, we can confidently say △GED≅△GFD\triangle GED \cong \triangle GFD△GED≅△GFD!

STEP 9

We have successfully proven that △GED≅△GFD\triangle GED \cong \triangle GFD△GED≅△GFD using the Hypotenuse-Leg Theorem!

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.

Questionse the given information to prove that $\triangle G E D \cong \triangle G F D$ .
Prove: $\triangle G E D \cong \triangle G F D$ Send To Proof Statement Reason
1 $\square$ Reason? Save For Later

What is this asking? We need to show that two triangles, GED and GFD, are congruent, knowing they share a side and have congruent sides DE and DF. Watch out! Don't assume anything beyond what's given!
We only know those two sides are equal, and the shared side.

1. State the givens
2. Establish shared side
3. Use Hypotenuse-Leg Theorem

We are given that $DE \cong DF$ .
This means the lengths of sides $DE$ and $DF$ are equal.
This is a key fact we'll use later!

We're also told that triangles GED and GFD are right triangles.
This means they each have a right angle, specifically, angles GED and GFD are right angles.

The triangles share the side $GD$ .
Because it's the same segment in both triangles, we know $GD \cong GD$ .
This is called the reflexive property!
We now have two pairs of congruent sides between the triangles!

In right triangles GED and GFD, sides GE and GF are opposite the right angles.
This makes GE and GF the hypotenuses of their respective triangles.

Therefore, by the HL Theorem, we can confidently say $\triangle GED \cong \triangle GFD$ !

We have successfully proven that $\triangle GED \cong \triangle GFD$ using the Hypotenuse-Leg Theorem!