Select All Transformations That Must Result in a Congruent Image
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.
Respective Vertices | A corresponds with D B corresponds with E C corresponds with F |
Corresponding Sides | AB = DE BC = EF AC = DF |
Corresponding Angles | thousand∠A = m∠D 1000∠B = m∠E grand∠C = m∠F |
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.
Respective Vertices | G corresponds with J H corresponds with One thousand I corresponds with L |
Corresponding Sides | GH = 10; JK = x; GH = JK Hello = 10; KL = ten; Hello = KL GI = 10√2; JL = 10√ii; GI = JL |
Corresponding Angles | g∠One thousand = 45°; m∠J = 45°; 1000∠G = m∠J m∠H = 90°; g∠K = 90°; chiliad∠H = m∠K thou∠I = 45°; m∠L = 45°; 1000∠I = yard∠L |
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.
Corresponding Vertices | M corresponds with R North corresponds with P O corresponds with Q |
Corresponding Sides | MN = 15; RP = 15; MN = RP NO = 12.1; PQ = 12.1; NO = PQ MO = 13.viii; RQ = 13.8; MO = RQ |
Corresponding Angles | yard∠M = 49.4°; m∠R = 49.4°; m∠One thousand = grand∠R m∠Northward = 60.26°; thou∠P = 60.26°; m∠N = yard∠P m∠O = 70.35°; m∠Q = 70.35°; m∠O = m∠Q |
Example #3
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.
Corresponding Vertices | S corresponds with 5 T corresponds with West U corresponds with X |
Respective Sides | ST corresponds with VW TU corresponds with WX SU corresponds with VX |
Corresponding Angles | ∠S corresponds with ∠V ∠T corresponds with ∠W ∠U corresponds with ∠X |
Since they are congruent, and so the respective sides and angles must besides be equal in measure. Hence,
Equal Sides | ST = VW TU = WX SU = VX |
Equal Angles | grand∠S = m∠Five yard∠T = m∠W m∠U = m∠X |
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.
Corresponding Vertices | A corresponds with X B corresponds with Y C corresponds withW |
Corresponding sides | AB = 17; XY = 17; AB = XY BC = 12; YW = 12; BC = YW Air conditioning = 21; XW = 21; AC = XW |
Since the AB = XY, BC = YW, and AC = XW, so we can say that △ABC ≅ △XYW by SSS Triangle Congruence Postulate. |
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
Pace-past-step Process | Explanation |
B corresponds with L A corresponds with E Thousand corresponds with T |
Assume the corresponding vertices of △Purse and △LET. |
BA corresponds with LE AG corresponds with ET BG corresponds with LT |
Presume the corresponding sides of △BAG and △Permit. |
BA = 16; LE = 13; BA ≠ LE AG = 13; ET = 16; AG ≠ ET BG = 10; LT = 10; BG ≠ LT |
Compare the side measure of each segments. |
Therefore, △Handbag and △Permit are not coinciding since not all three corresponding sides are of equal mensurate. |
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.
Case #one
Determine if the two triangles are congruent.
Solution
Identify all the corresponding vertices, sides, and angles.
Corresponding Vertices | B corresponds with G N corresponds with O East corresponds with T |
Corresponding Sides | BN = 10; GO = x; BN = Go NE = 13; OT = 13; NE = OT |
Corresponding Angle | m∠North = 75°; m∠O = 75°; ∠North ≅ ∠O |
Since the BN = Go, NE = OT, and the included bending betwixt them ∠North ≅ ∠O, then we can say that △BNE ≅ △GOT by SAS Triangle Congruence Postulate. |
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
Step-by-step Process | Caption |
H corresponds with S O corresponds with I P corresponds with T |
Assume the respective vertices of △HOP and △SIT. |
HO corresponds with SI OP corresponds with Information technology HP corresponds with ST |
Assume the corresponding sides of △HOP and △SIT. |
HO = 25; SI = 25; HO = SI OP = 16; Information technology = xvi; OP = IT |
Compare the side measure of each segments. |
m∠O = 65°; m∠I = 65°; ∠O ≅ ∠I | Compare the bending betwixt the segments HO and OP, and SI and IT. |
Therefore, by SAS Triangle Congruence Postulate, △HOP and △Sit down are coinciding since the two sides and the included angle are congruent with the respective sides of the other triangle. |
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.
Example #1
Determine if the 2 triangles are congruent.
Solution
Identify all the corresponding vertices, sides, and angles.
Corresponding Vertices | T corresponds with P U corresponds with A G corresponds with R |
Corresponding Angles | m∠U = 55°; m∠A = 55°; ∠U ≅ ∠A thou∠1000 = 82°; m∠R = 82°; ∠G ≅ ∠R |
Corresponding Side | UG = 21; AR = 21; UG = AR |
Since 2 angles and an included side are congruent, then by ASA Triangle Congruence Postulate, △TUG ≅ △PAR. |
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
Step-by-step Process | Caption |
B corresponds with W L corresponds with Y K corresponds with T |
Assume the corresponding vertices of BLK and WYT. |
B corresponds with Due west L corresponds with Y Chiliad corresponds with T |
Assume the corresponding angles of BLK and WYT. |
BL corresponds with WY LK corresponds with YT BK corresponds with WT |
Assume the corresponding sides of BLK and WYT. |
m∠L = 180° – (chiliad∠B + grand∠1000) yard∠L = 180° – (30° + 100°) m∠L = 180° – 130° m∠50 = fifty° |
Find the measure of the unknown bending L. |
one thousand∠T = 180° – (m∠W + m∠Y) g∠T = 180° – (30° + 100°) m∠T = 180° – 130° yard∠T = l° |
Discover the measure out of the unknown angle T. |
m∠B = 30°; m∠West = thirty°; ∠B ≅ ∠Due west m∠L = l°; thousand∠Y = 100°; one thousand∠L ≠ m∠Y g∠K = 100°; chiliad∠T = l°; m∠K ≠ one thousand∠T |
Compare the bending measure of each corresponding angle. |
BK = xl; WT = 40; BK = WT | Compare the given side measure to its corresponding side. |
Therefore, we △BLK and △WYT are non congruent because even though both triangles have ane equal corresponding side, no two respective angles are of equal measure. Hence, we cannot say that that it is congruent by ASA Triangle Congruence Postulate. |
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
Stride-by-step Procedure | Explanation |
L corresponds with C E corresponds with A North corresponds with K |
Assume the corresponding vertices of △LEN and △CAM. |
L corresponds with C E corresponds with A N corresponds with M |
Assume the corresponding angles of △LEN and △CAM. |
LE corresponds with CA EN corresponds with AM LN corresponds with CM |
Assume the corresponding sides of △LEN and △CAM. |
m∠L = 95°; m∠C = 95°; ∠L ≅ ∠C m∠E = 20°; m∠A = 20°; ∠E ≅ ∠A |
Compare the angle measure out of each corresponding angle. |
EN = 6; AM = vi; EN = AM | Compare the side measure of the corresponding non-included side of the 2 triangles. |
Since the 2 corresponding angles of the ii triangles are congruent, and the non-included sides are as well coinciding. Then, past AAS Triangle Theorem, △LEN ≅ △CAM. |
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
Step-past-step Procedure | Caption |
Y corresponds with C West corresponds with A Ten corresponds with B |
Identify the corresponding vertices of △YWX and △CAB. |
∠Y corresponds with ∠C ∠W corresponds with ∠A ∠10 corresponds with ∠B |
Identify the corresponding angles of △YWX and △CAB. |
YW corresponds with CA WX corresponds with AB YX corresponds with CB |
Identify the corresponding sides of △YWX and △CAB. |
m∠Due west = 90°; k∠A = 90°; ∠Due west ≅ ∠A | Compare the angle measure of each corresponding angle. |
YW = 20; CA = twenty; YW = CA BC = 35; XY = 35; BC = XY |
Compare the side measure out of each respective side. |
Since we accept two right-angled triangles where the hypotenuse are of equal measure and the corresponding sides are too congruent, then we can say that △YWX and △CAB are congruent by RHS Triangle Theorem. |
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
Footstep-by-step Process | Caption |
K corresponds with Southward R corresponds with M C corresponds with J |
Assume the corresponding vertices of △KRC and △SMJ. |
∠1000 corresponds with ∠S ∠R corresponds with ∠M ∠C corresponds with ∠J |
Presume the corresponding angles of △KRC and △SMJ. |
KR corresponds with SM RC corresponds with MJ KC corresponds with SJ |
Assume the respective sides of △KRC and △SMJ. |
thou∠R = xc°; thou∠1000 = ninety°; ∠R ≅ ∠Grand | Compare the angle measure of each corresponding angle. |
RC = 15; MJ = 15; RC = MJ | Compare the hypotenuse of each triangle. |
KR = $\sqrt{RC^2- KC^2}$ KR = $\sqrt{xv^2- 10^two}$ KR = $\sqrt{225- 100}$ KR = $\sqrt{125}$ KR = 5$\sqrt{five}$ |
Find the measure out of the other segment of △KRC using the Pythagorean Theorem. |
SJ = $\sqrt{MJ^2- SM^2}$ SJ = $\sqrt{xv^2- 10^ii}$ SJ = $\sqrt{225- 100}$ SJ = $\sqrt{125}$ SJ = 5$\sqrt{5}$ |
Notice the measure of the other segment of △SMJ using the Pythagorean Theorem. |
KR = 5$\sqrt{5}$; SM = 10; KR ≠ SM KC = ten; SJ = five$\sqrt{5}$; KC ≠ SJ |
Compare the side measure of each corresponding side. |
Hence, even if both triangles are right-angled and accept the same hypotenuse measure, △KRC and △SMJ are not congruent since we need a hypotenuse and some other side that have the same mensurate. |
Congruent Triangles (and other figures)
This department will help your child to place the characteristics of congruent 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:
- Rotation (Plough): Turns a effigy effectually a fixed point.
- Reflection (Flip): Flip of figure over a line where a mirror image is created.
- 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:
- Rotation (Turn): Turns a figure around a stock-still signal.
- Reflection (Flip): Flip of figure over a line where a mirror image is created.
- 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?
Recommended Worksheets
Congruent Shapes (World Tourism Day) Themed Math Worksheets
Right Triangles (Halloween Themed) Math Worksheets
Spatial Skills: Triangles and Quadrilaterals (Winter Solstice Themed) Math Worksheets
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Select All Transformations That Must Result in a Congruent Image
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