Lesson 3 Can You Say It with Symbols? Solidify Understanding

Ready

Remember when you write a congruence statement such as $△ A B C ≅ △ F G H$ , the corresponding parts of the two triangles must be the parts that are congruent.

For instance, from the congruence statements above we would know the following: $∠ A ≅ ∠ F$ , $\overset{―}{A B} ≅ \overset{―}{F G}$ , $∠ B ≅ ∠ G$ , $\overset{―}{B C} ≅ \overset{―}{G H}$ .

The segments and angles in each problem are corresponding parts of two congruent triangles. Make a sketch of the two triangles and mark congruent parts. Then identify the congruence pattern that justifies your statement. Finally, write a congruence statement for each pair of triangles represented.

1.

$\overset{―}{M L} ≅ \overset{―}{Z J}$ , $\overset{―}{L R} ≅ \overset{―}{J B}$ , $∠ L ≅ ∠ J$

a.

Sketch of Triangles

b.

Congruence pattern

c.

Congruence statement

2.

$\overset{―}{W B} ≅ \overset{―}{Q R}$ , $\overset{―}{B P} ≅ \overset{―}{R S}$ , $\overset{―}{W P} ≅ \overset{―}{Q S}$

a.

Sketch of Triangles

b.

Congruence pattern

c.

Congruence statement

3.

$\overset{―}{C Y} ≅ \overset{―}{R P}$ , $∠ Y ≅ ∠ P$ , $∠ E ≅ ∠ B$

a.

Sketch of Triangles

b.

Congruence pattern

c.

Congruence statement

4.

$\overset{―}{B C} ≅ \overset{―}{J K}$ , $\overset{―}{B A} ≅ \overset{―}{J M}$ , $∠ B ≅ ∠ J$

a.

Sketch of Triangles

b.

Congruence pattern

c.

Congruence statement

Set

Olivia is studying $△ A B C$ and quadrilateral $A D E G$ as shown in the figure. Her teacher has told the class that quadrilateral $A D E G$ is a rectangle and that $\overset{―}{G E}$ bisects both segments $\overset{―}{B C}$ and $\overset{―}{A B}$ . The assignment is to prove $△ B G E ≅ △ E D C$ in a two-column proof.

Olivia starts to organize her thinking by writing what she knows and the reasons she knows it.

I know $\overset{―}{G E}$ bisects $\overset{―}{B C}$ and $\overset{―}{A B}$ because I was given that information.
I know that $\overset{―}{B E} ≅ \overset{―}{E C}$ and $\overset{―}{B G} ≅ \overset{―}{A G}$ by definition of bisect.
I know that $\overset{―}{A G} ≅ \overset{―}{D E}$ because they are opposite sides of a rectangle.
I know that $\overset{―}{B G} ≅ \overset{―}{D E}$ by the transitive property.

Olivia realizes that she already has two sets of corresponding sides that are congruent in $△ B G E$ and $△ E D C$ . Now she only needs to prove the third sides congruent to have SSS or maybe find some angles that are congruent, in order to have a different congruence pattern. She wonders if the rectangle can tell her more about the two triangles.

I know that all of the angles in $A D E G$ are right angles because $A D E G$ is a rectangle.
I know that $∠ G A D$ and $∠ B G E$ are both right angles because they form a linear pair.
I know that $∠ A D E$ and $∠ C D E$ are both right angles because they form a linear pair.

5.

Olivia is excited to have two sides and an angle. But she quickly notices that she has SSA, and she knows that is a pattern she can’t depend on. But right triangles are special because if you have two sides of any right triangle, the third side can be found by doing the Pythagorean Theorem. Olivia believes she has her proof idea and begins to write her proof using as many mathematical abbreviations (symbols) as she can. Help Olivia finish her proof.

Given: quadrilateral $A D E G$ is a rectangle, $\overset{―}{E D}$ bisects $\overset{―}{B C}$

Prove: $Δ B G E ≅ Δ E D C$

Statements	Reasons
1. quadrilateral $A D E G$ is a rectangle	given
2. $\overset{―}{G E}$ bisects $\overset{―}{B C}$ and $\overset{―}{A B}$	given

Go

6.

Perform the following transformations on $△ A B C$ . Use a straightedge to connect the corresponding points with a line segment. Answer the questions.

Reflect $△ A B C$ over $\overset{―}{L K}$ . Label your new image $△ A^{'} B^{'} C^{'}$ .
What do you notice about the line segments $\overset{―}{A A^{'}}$ , $\overset{―}{B B^{'}}$ , and $\overset{―}{C C^{'}}$ ?
Compare line segments $\overset{―}{A B}$ , $\overset{―}{B C}$ , and $\overset{―}{C A}$ to $\overset{―}{A^{'} B^{'}}$ , $\overset{―}{B^{'} C^{'}}$ , and $\overset{―}{C^{'} A^{'}}$ . What is the same and what is different about these segments?
Translate $△ A B C$ down $8$ units and right $10$ units. Label your new image $△ A " B " C "$ .
What do you notice about the line segments $\overset{―}{A A^{″}}$ , $\overset{―}{B B^{″}}$ , and $\overset{―}{C C^{″}}$ ?
Compare line segments $\overset{―}{A B}$ , $\overset{―}{B C}$ , and $\overset{―}{C A}$ to $\overset{―}{A^{″} B^{″}}$ , $\overset{―}{B^{″} C^{″}}$ , and $\overset{―}{C^{″} A^{″}}$ . What is the same and what is different about these segments?
Translate $△ A B C$ down $10$ units and reflect it over the $y$ -axis. Label your new image $△ A^{‴} B^{‴} C^{‴}$ .
What do you notice about the line segments $\overset{―}{A A^{‴}}$ , $\overset{―}{B B^{‴}}$ , and $\overset{―}{C C^{‴}}$ ?
Compare line segments $\overset{―}{A B}$ , $\overset{―}{B C}$ , and $\overset{―}{C A}$ to $\overset{―}{A^{‴} B^{‴}}$ , $\overset{―}{B^{‴} C^{‴}}$ , and $\overset{―}{C^{‴} A^{‴}}$ . What is the same and what is different about these segments?