# Select All Transformations That Must Result in a Congruent Image

## What are congruent triangles?

When all corresponding sides and respective angles of two triangles are of the same measure, then it is said to be coinciding triangles. Triangles can be moved, rotated, flipped, and turned to expect exactly the same. More then, they coincide with each other when they are moved in the aforementioned position. To evidence congruence between two triangles, we use the symbol ≅.

The figure shows 2 congruent △ABC and △DEF where all respective sides and corresponding angles are equal.

Working through the lesson below volition help your child to understand that congruent figures can be determined by a effigy rotation, reflection, or translation or any combination of the three. Information technology volition as well help them to place the types of transformations in a sequence.

## What are the corresponding parts of triangles?

When nosotros learn about the congruent triangle, nosotros hear the discussion CPCT.
CPCT
stands for “Corresponding Parts of Congruent Triangles.” Hence, if all parts of two triangles have the same measure, nosotros utilise the term CPCT.

## How to identify corresponding parts of triangles?

For united states of america to say that two triangles are congruent, nosotros need to identify all the corresponding parts of the triangle. And then, how do we do information technology?

Nosotros will listing the respective vertices, sides, and angles using the same figure.

Note that y’all cannot interchange the letters because they should e’er correspond with the other vertex.

Example #i

Identify the corresponding parts of the two triangles.

Given the figure and its side measure and bending measure, we will list the corresponding vertices, respective sides, and corresponding angles.

Example #2

Identify the corresponding parts of the ii triangles.

Given the effigy and its side mensurate and bending measure out, we volition listing the corresponding vertices, corresponding sides, and corresponding angles.

Example #3

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If △STU is congruent with △VWX, which parts of the two triangles are equal?

Solution

Since △STU ≅ △VWX, then, nosotros first need to identify the corresponding parts of the triangles.

Since they are congruent, and so the respective sides and angles must besides be equal in measure. Hence,

## What are the tests for congruence?

If two triangles are of the same size and shape, then they are said to exist congruent. However, we do not need to detect all three pairs of corresponding sides and all iii pairs of corresponding angles to say that they are coinciding. There are some criterion, tests, or postulates that nosotros tin easily follow to say that two triangles are indeed congruent.

### SSS Triangle Congruence Postulate

SSS Triangle Congruence Postulate means Side-Side-Side Triangle Congruence Postulate. When iii sides of 1 triangle are coinciding with the three corresponding sides of some other triangle, the two triangles are congruent under the SSS Triangle Congruence Postulate.

In the given figure, OG = CT, OD = CA, and DG = AT, then by SSS Triangle Congruence Postulate, △DOG ≅ △Deed.

Example #one

Determine if the two triangles are congruent.

Solution

Identify all the respective vertices and sides.

Example #2

Are two equilateral triangles congruent?

Caption

Yes. Past SSS Triangle Congruence Postulate, we can prove that ii equilateral triangles are congruent since past definition, all sides of an equilateral are equal. Hence, all three sides of an equilateral are equal to the three corresponding sides of the other equilateral triangle.

Example #three

If the measures of the side of △BAG are BA = sixteen, AG = xiii, BG = 10 and the side measure of another triangle are LE = 13, ET = 16, LT = 10. Tin we say that △Purse and △Allow are congruent past SSS Triangle Congruence Postulate?

Solution

### SAS Triangle Congruence Postulate

SAS Triangle Congruence Postulate means Side-Angle-Side Triangle Congruence Postulate. When two sides and an included angle of ane triangle are congruent to the corresponding two sides and an included angle of some other triangle, the two triangles are said to be congruent by SAS Triangle Congruence Postulate.

In the given figure, HT = JA, TU = AG, and the included angle ∠T ≅ ∠A, then by SAS Triangle Congruence Postulate, △HTU ≅ △JAG.

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Case #one

Determine if the two triangles are congruent.

Solution

Identify all the corresponding vertices, sides, and angles.

Example #2

The measure of the sides of △HOP are HO = 25 and OP = 16, and the angle measure out of the included angle is m∠O = 65°. Meanwhile, some other triangle △SIT has side measures of SI = 25 and Information technology = 16 and the measure of the included angle is yard∠I = 65°. Can we say that △HOP and △Sit down are congruent by SAS Triangle Congruence Postulate?

Solution

### ASA Triangle Congruence Postulate

ASA Triangle Congruence Postulate means Angle-Side-Bending Triangle Congruence Postulate. So, when two angles and the included side of 1 triangle are coinciding with the respective two angles and 1 included side of some other triangle, then the two triangles are congruent.

In the given figure, the angles ∠Y ≅ ∠T and ∠V ≅ ∠I and the included side betwixt them YV = TI. Hence, △YVE and △TIM are coinciding by ASA Triangle Congruence Postulate.

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Example #1

Determine if the 2 triangles are congruent.

Solution

Identify all the corresponding vertices, sides, and angles.

Example #2

The mensurate of the angles of △BLK are m∠B = 30° and m∠G = 100°, and the side in betwixt them measures BK = 40. Some other triangle, △WYT has angle measures of m∠W = xxx° and m∠Y = 100° and a side of WT = 40, is it true that △BLK is congruent with △WYT by ASA Triangle Congruence Postulate?

Solution

### AAS Triangle Theorem

AAS Triangle Theorem means Angle-Angle-Side Triangle Theorem. The AAS Triangle Theorem states that if the 2 angles and a not-included side of one triangle are congruent to the corresponding ii angles and a non-included side, and so the two triangles are said to be congruent.

Two triangles △PAT and △FGO are said to be coinciding since ∠T ≅ ∠O and ∠A ≅ ∠Yard and the AP = GF which is the respective non-included side are too coinciding with each other. Hence, past AAS Triangle Theorem, △PAT ≅ △FGO.

Example

The angle measures of △LEN are m∠50 = 95° and m∠Eastward = xx° where a side not between them EN measures vi centimeters. Meanwhile, △CAM has angles that measures chiliad∠C = 95° and m∠A = 20° with a not-included side that measures AM = half-dozen cm. Can we say that △LEN and △CAM are congruent?

Solution

### RHS Triangle Theorem

RHS Triangle Theorem ways Right-Hypotenuse-Side Triangle Theorem. In this theorem, information technology states that if the hypotenuse and the side of a correct triangle are congruent to the corresponding hypotenuse and the side of the other correct triangle, so the two right triangles are said to be congruent by RHS Triangle Theorem.

In the given right triangle △GWY and △LUN,  ∠G ≅ ∠L because they both mensurate 90°. More than then, the hypotenuse of △GWY and △LUN which is WY and Un are as well congruent, respectively. Lastly, the sides GY of △GWY and NL of △LUN are as well congruent. Hence, by RHS Triangle Theorem, △GWY ≅ △LUN.

Case #one

Determine if the ii triangles are congruent.

Solution

Example #ii

△KRC is a right triangle where the hypotenuse RC measures fifteen units and the other side KC measures 10 units. Meanwhile, another triangle △SMJ is likewise a right triangle with the aforementioned hypotenuse measure out as KRC, and SM measures 10 units. Is information technology true that △KRC and △SMJ are congruent by RHS Triangle Theorem?

Solution

## Congruent Triangles (and other figures)

Two figures are congruent if they are the:

• Exact same shape
• Exact aforementioned size
• Angle measures are equal
• Line segments are equal

Expect at the instance below.

Discuss the examples and questions below with your kid regarding whether the figures are coinciding.

Which figure is coinciding to figure C shown beneath?

Figure b. is congruent.

## Transformations : Rotations, Reflections, & Translations

This section will aid your kid to perform a transformation (rotation, reflection, and translation) on a figure .

Make sure your child is familiar with the vocabulary beneath:

• Transformation moves a figure from its original identify to a new place.
• Angle of Rotation:How big the angle is that yous rotate a figure.  Common angle rotations are 45°, xc°, 180°.
• Isometric Transformation:A transformation that does not change the size of a figure.

In that location are three types of transformations. Alternative names are in parenthesis:

1. Rotation (Plough): Turns a effigy effectually a fixed point.
2. Reflection (Flip): Flip of figure over a line where a mirror image is created.
3. Translation (Slide or glide): Sliding a shape to a new place without changing the effigy.

Rotations, reflections, and translations are isometric.  That means that these transformations do not change the size of the figure.  If the size and shape of the figure is non inverse, then the figures are congruent.

Explore and discuss the examples of transformations below with your kid.

Try Information technology! Find a flat object in your dwelling that can hands be moved (small book, calculator, drinkable coaster, money, etc.)  Perform each transformation using that object.

### Multiple Transformations

This department will help your child to understand that coinciding figures can have more than one transformation.

Make sure your child is familiar with the vocabulary below:

• Sequence: A group of things arranged in a certain society.  Usually known as a design.

Recapping from earlier in his lesson, there are three types of transformation:

1. Rotation (Turn): Turns a figure around a stock-still signal.
2. Reflection (Flip): Flip of figure over a line where a mirror image is created.
3. Translation (Slide or glide): Sliding a shape to a new place without irresolute the figure.

Two Transformations

Try It! Wait at the figure below. What transformations does parallelogram Z perform?

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## Select All Transformations That Must Result in a Congruent Image

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