5. Congruence and Triangles

5.02 Corresponding parts of congruent triangles

Lesson

Practice

5.03 SSS and SAS congruence criteria

5.04 ASA and AAS congruence criteria

5.05 Right triangle congruence

5.06 Modeling with congruent triangles (M)

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United States of America

Grades 9-12

5.02 Corresponding parts of congruent triangles

Lesson

Practice

Introduction

We have been building on the concept of congruency from 8th grade through Geometry. In lesson 5.01 Congruence transformations , we concluded that two figures are congruent if and only if a rigid transformation exists that maps one figure onto the other. We will extend that concept to a theorem that we can use to justify congruency between triangles.

Ideas

Corresponding parts of congruent triangles

Corresponding parts of congruent triangles

Exploration

Consider the diagram below. \triangle{ABC} \cong \triangle{DEF}.

Triangle A B C and D E F. Speak to your teacher for more details.

What sequence of rigid transformations could map \triangle{ABC} to \triangle{DEF}?
What segment is the image of \overline{AC} after performing rigid motions on the pre-image? How do you know?
What angle is the image of \angle B after performing motions on the pre-image? How do you know?
Each segment and each angle in \triangle{ABC} maps to another segment or angle in \triangle{DEF}. What can you say about the relationship between each mapped pair?

When a figure can be mapped to an image using translations, reflections, and rotations, we can state that the triangles are congruent by definition of rigid motion transformations. Once we've established congruency between two triangles by mapping transformations, we can then justify the congruence of any pair of corresponding parts.

Rigid transformation

A transformation of points in space consisting of a sequence of one or more translations, reflections, or rotations. Rigid transformations preserve distances and angle measures (congruency).

Corresponding parts

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

Corresponding parts of congruent triangles theorem

Corresponding parts of congruent triangles are congruent

Two triangles. The 3 sides and 3 angles of one triangle are congruent to the other triangle.

We can reference this theorem with its acronym, CPCTC.

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 us 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}

Likewise, if we know all corresponding parts of a figure are congruent, then we know there is a way to map one figure to the other. This means two figures are congruent if all corresponding sides and all corresponding angles are congruent.

Examples

Example 1

Prove corresponding segments of congruent triangles are congruent.

Worked Solution

Create a strategy

Draw two triangles such that \triangle ABC \cong \triangle PQR.

We can use what we know about rigid transformations in this proof.

Apply the idea

If figures are congruent, there is a rigid transformation that maps one figure onto the other
If there is a rigid transformation that maps one figure onto another, it also maps one segment of the figure onto a segment of the other, and we call these two corresponding segments
Since there is a rigid transformation mapping one segment onto another, those segments are congruent
Therefore, if figures are congruent, then the corresponding segments of those figures must also be congruent

Reflect and check

We could use the same steps to prove corresponding angles are congruent:

If figures are congruent, there is a rigid transformation that maps one figure onto the other
If there is a rigid transformation that maps one figure onto another, it also maps one angle of the figure onto an angle of the other, and we call these two corresponding angles
Since there is a rigid transformation mapping one angle onto another, those angles are congruent
Therefore, if figures are congruent, then the corresponding angles of those figures must also be congruent

Example 2

Consider the congruent triangles shown below:

Show that the triangles are congruent using rigid motions.

Worked Solution

Apply the idea

A reflection across a horizontal line, a rotation 40 \degree counterclockwise about R, and a translation along a vector from R'' to K will map one triangle onto the other, as shown:

A diagram showing how triangle Q R S is reflected across a horizontal line. Speak to your teacher for more details.

A diagram showing how triangle Q R S is rotated 40 degrees counterclockwise after reflection. Speak to your teacher for more details.

A diagram showing how triangle Q R S is translated after reflection and rotation. Triangle J K L is the resulting triangle after the transformations. Speak to your teacher for more details.

Reflect and check

The rigid motions required to map one triangle onto the other can be performed in another order, such as a translation, followed by a reflection, followed by a rotation in this case.

Note that the order of transformations cannot always be changed for mapping one figure onto another.

Identify the congruent parts of the triangles and write a congruency statement.

Worked Solution

Create a strategy

The corresponding parts of the triangles are marked. We can list their congruency, then write an overall congruency statement for the triangles.

Apply the idea

\overline{JK} \cong \overline{QR}
\overline{KL} \cong \overline{RS}
\overline{LJ} \cong \overline{SQ}
\angle J \cong \angle Q
\angle K \cong \angle R
\angle L \cong \angle S

\triangle{JKL} \cong \triangle{QRS}

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 3 y plus 23. Adjacent to it is side of length 5 x minus 20. The second triangle has side of length 5 y minus 12 opposite the marked angle. Adjacent to this angle is side of length 2 x plus 13.

Worked Solution

Create a strategy

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 with the y expression. Those sides must also be congruent.

Set the expressions equal to each other and solve for the unknown variables.

Apply the idea

We have:

\displaystyle 5x-20	\displaystyle =	\displaystyle 2x+13	Congruent sides
\displaystyle 5x	\displaystyle =	\displaystyle 2x+33	Add 20 to both sides
\displaystyle 3x	\displaystyle =	\displaystyle 33	Subtract 2x from both sides
\displaystyle x	\displaystyle =	\displaystyle 11	Divide by 3 on both sides

Next, we have:

\displaystyle 3y+23	\displaystyle =	\displaystyle 5y-12	Congruent sides
\displaystyle 23	\displaystyle =	\displaystyle 2y-12	Subtract 2y from both sides
\displaystyle 35	\displaystyle =	\displaystyle 2y	Add 12 to both sides
\displaystyle 17.5	\displaystyle =	\displaystyle y	Divide by 2 on both sides

Reflect and check

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

We can also confirm that relative side lengths should still match up proportionally: the shortest side of one triangle will be the same length as the shortest side of the other.

\displaystyle 5x-20	\displaystyle =	\displaystyle 5(11)-20	Substitute x=11
	\displaystyle =	\displaystyle 35	Evaluate the multiplication and addition

\displaystyle 2x+13	\displaystyle =	\displaystyle 2(11)+13	Substitute x=11
	\displaystyle =	\displaystyle 35	Evaluate the multiplication and addition

Example 4

Hamida is working on a building for her architecture class final exam. A tower in her building is constructed from congruent triangles that meet at the top of the tower's roof.

A figure of a tower's roof. Three triangles are drawn on the outline of the roof: A B C, C B D, and D B E. Vertex B is at the apex of the roof. Speak to your teacher for more details.

If \angle BCD \cong \angle BDC, and m \angle BCD=67 \degree, determine m \angle DBE.

Worked Solution

Create a strategy

We know that \triangle BCD \cong \triangle BDE, so we will use what we know about \triangle BCD to solve unknown parts of \triangle BDE.

Apply the idea

\displaystyle m \angle BCD	\displaystyle =	\displaystyle m \angle BDC	Definition of congruence
\displaystyle 67	\displaystyle =	\displaystyle m \angle BDC	Substitution
\displaystyle m \angle BCD + m \angle BDC + m \angle CBD	\displaystyle =	\displaystyle 180	Triangle sum theorem
\displaystyle 67 + 67 + m \angle CBD	\displaystyle =	\displaystyle 180	Substitution
\displaystyle 134 + m \angle CBD	\displaystyle =	\displaystyle 180	Combine like terms
\displaystyle m \angle CBD	\displaystyle =	\displaystyle 46	Subtract 134 from both sides
\displaystyle m \angle CBD	\displaystyle =	\displaystyle m \angle DBE	CPCTC and definition of congruence
\displaystyle 46	\displaystyle =	\displaystyle m \angle DBE	Substitution

m \angle DBE = 46 \degree.

Idea summary

If two triangles are congruent, all corresponding segment and angle pairs will be congruent.

If all corresponding sides and all corresponding angles between a pair of figures are congruent, then the two figures will be congruent.

Outcomes

G.CO.B.7

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.

5.02 Corresponding parts of congruent triangles

Introduction

Ideas

Corresponding parts of congruent triangles

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

G.CO.B.7

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