Congruent Triangles 1

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Geometry

Similarity and Congruence

Congruent Triangles

Congruent Triangles 1

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Question 1 of 4

1. Question
Is $∆XYU$ congruent to $∆ZUW$
- 1. No
- 2. Not enough information to say
- 3. Yes
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Two triangles are congruent if all corresponding angles and sides are equal

More Info

Use the given diagram to determine whether any of the angles or sides are congruent

Statement: $XU$ $=$ $UW$

Reason: Given by the markings (two dashes) on each segment

$(S)$ Side

Statement: $/_ X$ $=$ $/_ W$

Reason: Alternate Angles are equal

$(A)$ Angle

Statement: $/_ XUY$ $=$ $/_ ZUW$

Reason: Vertically Opposite angles are equal

$(A)$ Angle

We have proved congruence of $1$ side and $2$ angles between the two triangles. Therefore, the two triangles are congruent according to the Angle-Angle-Side $\text((AAS))$ criteria

Yes, $∆XYU$ is congruent to $∆ZUW$

More Info

Congruent Triangles

Use any of the following 4 tests to prove congruence between two triangles:

Side-Side-Side $\text((SSS))$

If 3 sides of one triangle are equal to 3 sides of another triangle, then the triangles are congruent.

Side-Angle-Side $\text((SAS))$

Two triangles are congruent if 2 sides and the included angle of one are respectively equal to 2 sides and the included angle of the other

Angle-Angle-Side $\text((AAS))$

Two triangles are congruent if 2 angles and a side of one are respectively equal to 2 angles and the corresponding side of the other.

Right Angle-Hypotenuse-Side $\text((RHS))$

Two right angled triangles are congruent if the hypotenuse and a side of one are respectively equal to the hypotenuse and a side of another

Alternate, Corresponding and Co-Interior Angle Types

Alternate Angles

Alternate Angles in parallel lines are equal. These angles typically form a Z shape.

Corresponding Angles

Corresponding Angles in parallel lines are equal. These angles typically form an F shape.

Co-Interior Angles

Co-Interior Angles are when two angles have a sum of $180°.$ These angles typically form a C Shape
Question 2 of 4

2. Question
Is $∆XYW$ congruent to $∆YZW$
- 1. Not enough information to say
- 2. No
- 3. Yes
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Two triangles are congruent if all corresponding angles and sides are equal

More Info

Use the given diagram to determine whether any of the angles or sides are congruent

Statement: $XY$ $=$ $YZ$

Reason: Given by the markings (two dashes) on each segment

$(S)$ Side

Statement: $XW$ $=$ $WZ$

Reason: Given by the markings (one dash) on each segment

$(S)$ Side

Statement: $YW$ $=$ $YW$

Reason: This is a common side for both triangles

$(S)$ Side

We have proved congruence of $3$ sides angles between the two triangles. Therefore, the two triangles are congruent according to the Side-Side-Side $\text((SSS))$ criteria

Yes, $∆XYW$ is congruent to $∆YZW$

More Info

Congruent Triangles

Use any of the following 4 tests to prove congruence between two triangles:

Side-Side-Side $\text((SSS))$

If 3 sides of one triangle are equal to 3 sides of another triangle, then the triangles are congruent.

Side-Angle-Side $\text((SAS))$

Two triangles are congruent if 2 sides and the included angle of one are respectively equal to 2 sides and the included angle of the other

Angle-Angle-Side $\text((AAS))$

Two triangles are congruent if 2 angles and a side of one are respectively equal to 2 angles and the corresponding side of the other.

Right Angle-Hypotenuse-Side $\text((RHS))$

Two right angled triangles are congruent if the hypotenuse and a side of one are respectively equal to the hypotenuse and a side of another

Alternate, Corresponding and Co-Interior Angle Types

Alternate Angles

Alternate Angles in parallel lines are equal. These angles typically form a Z shape.

Corresponding Angles

Corresponding Angles in parallel lines are equal. These angles typically form an F shape.

Co-Interior Angles

Co-Interior Angles are when two angles have a sum of $180°.$ These angles typically form a C Shape
Question 3 of 4

3. Question
Is $∆AED$ congruent to $∆EBC$
- 1. Yes
- 2. No
- 3. Not enough information to say
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Two triangles are congruent if all corresponding angles and sides are equal

More Info

Use the given diagram to determine whether any of the angles or sides are congruent

Statement: $AE$ $=$ $EC$

Reason: Given by the markings (two dashes) on each segment

$(S)$ Side

Statement: $/_ AED$ $=$ $/_ BEC$

Reason: Vertically Opposite angles are equal

$(A)$ Angle

Statement: $DE$ $=$ $EB$

Reason: Given by the markings (one dash) on each segment

$(S)$ Side

We have proved congruence of $2$ sides and $1$ angle between the two triangles. Therefore, the two triangles are congruent according to the Side-Angle-Side $\text((SAS))$ criteria

Yes, $∆AED$ is congruent to $∆EBC$

More Info

Congruent Triangles

Use any of the following 4 tests to prove congruence between two triangles:

Side-Side-Side $\text((SSS))$

If 3 sides of one triangle are equal to 3 sides of another triangle, then the triangles are congruent.

Side-Angle-Side $\text((SAS))$

Two triangles are congruent if 2 sides and the included angle of one are respectively equal to 2 sides and the included angle of the other

Angle-Angle-Side $\text((AAS))$

Two triangles are congruent if 2 angles and a side of one are respectively equal to 2 angles and the corresponding side of the other.

Right Angle-Hypotenuse-Side $\text((RHS))$

Two right angled triangles are congruent if the hypotenuse and a side of one are respectively equal to the hypotenuse and a side of another

Alternate, Corresponding and Co-Interior Angle Types

Alternate Angles

Alternate Angles in parallel lines are equal. These angles typically form a Z shape.

Corresponding Angles

Corresponding Angles in parallel lines are equal. These angles typically form an F shape.

Co-Interior Angles

Co-Interior Angles are when two angles have a sum of $180°.$ These angles typically form a C Shape
Question 4 of 4

4. Question
Is side $BC$ equal to side $EF$
- 1. Not enough information to say
- 2. No
- 3. Yes
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Two triangles are congruent if all corresponding angles and sides are equal

More Info

Use the given diagram to determine whether any of the angles or sides are congruent

Statement: $/_ A$ $=$ $/_ D$

Reason: Given by the values ( $55°$ ) on each segment

$(A)$ Angle

Statement: $/_ C$ $=$ $/_ F$

Reason: Given by the values ( $65°$ ) on each segment

$(A)$ Angle

Statement: $AB$ $=$ $ED$

Reason: Given by the markings (one dash) on each segment

$(S)$ Side

We have proved congruence of $2$ angles and $1$ side between the two triangles. Therefore, the two triangles are congruent according to the Angle-Angle-Side $\text((AAS))$ criteria

Since we have proven the congruence of the two triangles, it means that corresponding sides, such as sides $BC$ and $EF$ , are equal.

Yes, side $BC$ is equal to side $EF$

More Info

Congruent Triangles

Use any of the following 4 tests to prove congruence between two triangles:

Side-Side-Side $\text((SSS))$

If 3 sides of one triangle are equal to 3 sides of another triangle, then the triangles are congruent.

Side-Angle-Side $\text((SAS))$

Two triangles are congruent if 2 sides and the included angle of one are respectively equal to 2 sides and the included angle of the other

Angle-Angle-Side $\text((AAS))$

Two triangles are congruent if 2 angles and a side of one are respectively equal to 2 angles and the corresponding side of the other.

Right Angle-Hypotenuse-Side $\text((RHS))$

Two right angled triangles are congruent if the hypotenuse and a side of one are respectively equal to the hypotenuse and a side of another

Alternate, Corresponding and Co-Interior Angle Types

Alternate Angles

Alternate Angles in parallel lines are equal. These angles typically form a Z shape.

Corresponding Angles

Corresponding Angles in parallel lines are equal. These angles typically form an F shape.

Co-Interior Angles

Co-Interior Angles are when two angles have a sum of $180°.$ These angles typically form a C Shape

Congruent Triangles 1

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Quiz summary

Information

1. Question

Hint

Well Done!

Side-Side-Side (SSS)\text((SSS))

Side-Angle-Side (SAS)\text((SAS))

Angle-Angle-Side (AAS)\text((AAS))

Right Angle-Hypotenuse-Side (RHS)\text((RHS))

Alternate Angles

Corresponding Angles

Co-Interior Angles

2. Question

Hint

Nice Job!

Side-Side-Side (SSS)\text((SSS))

Side-Angle-Side (SAS)\text((SAS))

Angle-Angle-Side (AAS)\text((AAS))

Right Angle-Hypotenuse-Side (RHS)\text((RHS))

Alternate Angles

Corresponding Angles

Co-Interior Angles

3. Question

Hint

Excellent!

Side-Side-Side (SSS)\text((SSS))

Side-Angle-Side (SAS)\text((SAS))

Angle-Angle-Side (AAS)\text((AAS))

Right Angle-Hypotenuse-Side (RHS)\text((RHS))

Alternate Angles

Corresponding Angles

Co-Interior Angles

4. Question

Hint

Fantastic!

Side-Side-Side (SSS)\text((SSS))

Side-Angle-Side (SAS)\text((SAS))

Angle-Angle-Side (AAS)\text((AAS))

Right Angle-Hypotenuse-Side (RHS)\text((RHS))

Alternate Angles

Corresponding Angles

Co-Interior Angles

Quizzes

Side-Side-Side $\text((SSS))$

Side-Angle-Side $\text((SAS))$

Angle-Angle-Side $\text((AAS))$

Right Angle-Hypotenuse-Side $\text((RHS))$

Side-Side-Side $\text((SSS))$

Side-Angle-Side $\text((SAS))$

Angle-Angle-Side $\text((AAS))$

Right Angle-Hypotenuse-Side $\text((RHS))$

Side-Side-Side $\text((SSS))$

Side-Angle-Side $\text((SAS))$

Angle-Angle-Side $\text((AAS))$

Right Angle-Hypotenuse-Side $\text((RHS))$

Side-Side-Side $\text((SSS))$

Side-Angle-Side $\text((SAS))$

Angle-Angle-Side $\text((AAS))$

Right Angle-Hypotenuse-Side $\text((RHS))$