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4. Congruence & Triangles
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United States of AmericaTennessee
Geometry

4.01 Congruent figures

Lesson
Practice
Lesson

Concept summary

Two figures are congruent if all corresponding sides and all corresponding angles are congruent.

Congruent

Having exactly the same shape and size

Corresponding parts

Parts in congruent figures that occupy the same relative location in the figures.

When the figures are oriented in the same direction, it is easier to identify the corresponding parts. If the figures have been reflected or rotated try to find a reference point (such as a labeled pair, a shared side, or a right angle) to help you identify the corresponding parts.

A Coordinate plane with two triangles shown in Quadrant 1 and Quadrant 2. Triangle A B C with right angle B is plotted in quadrant 2 with point A at (negative 3, 4), Point B at (negative 3,1) and Point C at (negative 1, 1). Triangle X Y Z is plotted in Quadrant 1 with point X at (1,4), point Y at (1,1) and point Z at (3,1). Side A C and side X Y are marked congruent, as well as side B C and side Y Z, also side A B and side X Y. Angle B A C is marked congruent with Angle Y X Z, as well as Angle A C B and X Z Y

These figures are congruent by translation. The corresponding parts are all congruent:

\angle{A}\cong\angle{X}

\angle{B}\cong\angle{Y}

\angle{C}\cong\angle{Z}

\overline{AB}\cong\overline{XY}

\overline{BC}\cong\overline{YZ}

\overline{AC}\cong\overline{XZ}

Another way to prove or show that two figures are congruent is to provide a sequence of rigid transformations that map one figure onto the other. Since rigid transformations preserve side length and angle measure, any combination of translations, reflections, and rotations will always produce congruent figures.

Two triangles, triangle A B C and triangle D E F are drawn. Side A C and side D F are marked congruent as well as side A B and side E D, side C B and side E F. Angle A C B and angle D F E are marked congruent as well as angle C A B and E D F, and angle A B C and E F G.
These triangles are congruent because all sides and all angles are labeled as congruent
A four quadrant coordinate plane with two triangles plotted at Quadrants 1 and 4. Triangle A B C is plotted with Point A at ( negative 3, 4), point B at ( negative 3, 1) and point C at ( negative 1, 1). Triangle D E F is plotted at quadrant 4 with point D at (1, negative 1), E at (3, negative 1) and F at (1, negative 4).
\triangle{ABC}\cong\triangle{FDE} because a translation and reflection will map them onto each other

Worked examples

Example 1

The triangles in the diagram are congruent.

Triangle A B C and triangle P Q R are drawn such that segment A B and segment P R are marked congruent, as well as segment B C and segment Q R. Segment A C and segment P Q are also marked congruent.
a

Identify the transformations that proves these triangles are congruent.

Approach

There are three rigid transformations that preserve length and angle measure: translations, reflections, and rotations. Which transformations can be applied to map one triangle onto the other?

Solution

A reflection and a translation

Reflection

It looks like a rotation would map the triangles onto each other, but after rotating one triangle the congruency marks do not correspond correctly.

b

Complete the congruency statement: \triangle{ABC}\cong \triangle{⬚}

Approach

Identify the vertices that correspond to each A, B, C, then put them in the same order.

Solution

A is opposite the side with two congruent marks and so A corresponds to P. B is opposite the side with three congruent marks so B corresponds to R and C is opposite the side with one congruent mark so C corresponds to Q.

\triangle{ABC}\cong \triangle{PRQ}

Reflection

The order of a congruency statement matters. Different orders give different correspondences and indicate different sides and angles are congruent.

Example 2

Given that these two triangles are congruent, find the angle which corresponds to \angle{BCA}.

Triangle A B C and triangle O P Q are drawn such that segment A B and segment O P are marked congruent, as well as segment B C and segment Q P, also segment A C and segment O Q.

Approach

Since these triangles are not oriented in the same way, we need to find some other point of relativity to compare them. We can see that \angle{BCA} is opposite side \overline{BA} and that \overline{BA}\cong\overline{OP}. The angle opposite \overline{OP} is the angle corresponding to \angle{BCA}.

Solution

\angle{PQO} or \angle{OQP} corresponds to \angle{BCA}.

Reflection

We can also consider that \angle{BCA} is in between the sides with one and three congruent marks. This means that \angle{BCA} must correspond to \angle{PQO} which is also between the sides with one and three congruent marks.

Example 3

Given that these shapes are congruent, find the values of x and y.

Two congruent triangles are shown each with an angle marked with an arc. For the first triangle, opposite the marked angle is side of length 13. Adjacent to it is side of length 7. The second triangle has side y opposite the marked angle. Adjacent to this angle is side labelled x.

Approach

We are given a pair of corresponding, congruent angles in the diagram. Start with the sides opposite those angles as these sides must also be congruent. Next, look at the side adjacent to the angle in the direction of the side labeled y. Those sides must also be congruent.

Two triangles, each with an angle marked with an arc. Corresponding sides are color coded. For the first triangle, opposite the angle is side of length 13 which corresponds to the side y of the second triangle. Adjacent to the angle of the first triangle is side of length 7 which corresponds to x, the side adjacent to the angle in the direction of the side labelled y.

Solution

x=7 and y=13

Reflection

Not all pictures are drawn to scale. When determining what sides correspond, use a reference point based on the given congruences.

Outcomes

G.CO.B.5

Given two figures, use the definition of congruence in terms of rigid motions to determine informally if they are congruent.

G.CO.B.6

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G.MP1

Make sense of problems and persevere in solving them.

G.MP2

Reason abstractly and quantitatively.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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