Show:
Congruency
Angles
Transformations
Triangles
Bonus Challenge
100

True or False:

Two squares with the same side lengths are always congruent.

Show your reasoning. 

True. Congruent figures must have the exact same shape and size. Every square inherently has four 90 degree angles, its geometric shape is completely fixed. 

100

1. Find the measure of angle JGH. Explain or show your reasoning.

2.  Find and label a second 30 degree angle in the diagram. Find and label an angle congruent to angle JGH.

3.  Angle BAC is a right angle. Find the measure of angle CAD .


1. 150 degrees

2.


3. 26 degrees since 90 - 64= 26

100

Select all of the translations that take Triangle T to Triangle U. There may be more than one correct answer.

A. Translate  (-3,0) to (1,2).

B. Translate (2,1) to (-2, -1).

C. Translate (-4, -3) to (0, -1).

D. Translate  (1, 2) to (2, 1).

A and C both take Triangle T to Triangle U.

100

Examine this set of triangles.

  1. What is the same about the triangles in the set? What is different?

  2. How many different triangles are there? Explain or show your reasoning.

  1. All of the triangles have one side of length 5 and one side of length 7. And all of them have a 30-degree angle.The triangles are oriented differently, and the two sides and one angle are in a different order.
  2. There are 4 different triangles.
100

What is a sequence of transformation?

sequence of transformations

A sequence of transformations is a set of translations, rotations, reflections, and dilations on a figure. The transformations are performed in a given order.

This diagram shows a sequence of transformations to move Figure A to Figure C. 

First, A is translated to the right to make B. Next, B is reflected across line L to make C.


200

True or False.

Two rectangles with the same side lengths are always congruent. Explain your reasoning.

True.

Equal Side Lengths: A rectangle has two pairs of equal opposite sides.  All four corresponding sides are equal. 

Fixed Angles: By definition, every rectangle has four interior right angles measuring 90 degrees. 

200

  1. Find the measures of angles q, r, and s.
  2. Find the sum of the measures of angles q, r, and s.

  3. What do you notice about these three angles?


  1. q = 100, r = 135, s = 125 
  2. q + r + s = 360
  3. Sample response: The three angles together make one full revolution of a circle, or 360 degrees.
200

We can reflect triangle  ABC across side AC to form a new triangle:



200

Tyler has 3 right angles. Can he use them to make a triangle? Explain your reasoning.

No. Sample reasoning: 3 right angles sums to more than 180 degrees, since 3 * 90 = 270.

200

Lines AC and DF are parallel. They are cut by transversal HJ.

Find the seven unknown angle measures in the diagram.

The three 117-degree angles. Each of the other four angles is supplementary to a 117-degree angle, so they are all 63-degree angles.

300

True or False.

Two rhombuses with the same side lengths are always congruent. Explain your reasoning.

False. 

While all four sides of both rhombuses will be equal in length, their angles can be completely different. For two geometric shapes to be congruent, they must have the exact same shape and size, meaning both their corresponding sides and corresponding angles must be equal.





300

Lines AB and CD are parallel. Find the measures of the three angles in triangle ABF.

B: 42 degrees

A: 23 degrees

F: 115 degrees

300

For each pair of shapes, decide whether or not Shape A is congruent to Shape B. Explain your reasoning.


Not congruent. The shapes look congruent, but when Shape A is moved on top of Shape B with a 90-degree counterclockwise rotation and a translation, they do not match up.

300

Describe a way to move triangle ABC so that it matches up perfectly with triangle FED .


Triangle ABC can be moved down 2 units and then flipped over a vertical line that lies halfway between the 2 triangles.

300

What do you notice?

translation to take P to Q or vice versa. 

vertical angles are congruent

what rigid transformation tells you that vertical angles are congruent?

400

True or False.

Two parallelograms with the same side lengths are always congruent. Explain your reasoning with words and models. 

False.

Angles Can Vary.

Different Shapes.

Draw it! 

400

Find supplementary, complementary, and/or vertical angles. 

Angles a and d are vertical angles.

Angles a and e are supplementary angles.

400

For each pair of shapes, decide whether or not Shape A is congruent to Shape B. Explain your reasoning.


Congruent. If Shape A is reflected over its right side, then rotated 90 degrees counterclockwise around the lower vertex, it can be placed on top of Shape B with a translation down and to the right.

400

Draw as many different triangles as possible that have two sides of length 4 cm and a  angle. Clearly mark the side lengths and angles given.


400

What is a tesselation?

tessellation


A tessellation is a repeating pattern of 1 or more shapes. The sides of the shapes fit together with no gaps or overlaps. The pattern goes on forever in all directions.

This diagram shows part of a tessellation.


500

True or false. 

Two quadrilaterals with the same side lengths are always congruent. Explain and draw your reasoning. 

Fasle. 

Triangles are rigid: Unlike triangles—which are strictly determined by their three side lengths.

Flexibility of angles: You can "squish" or deform a four-sided frame (like a square or a rhombus) into many different shapes with the exact same side lengths but completely different interior angles and diagonal lengths.

500

1. Find the angles with a sum of 180 degrees. 

2. Find angles with equal measurements.

1. CBD and CBA

2. CBA, DBE -  CBD, ABE


500

Describe a sequence of transformations that shows that Polygon A is congruent to Polygon B.


Polygon A can be rotated 90 degrees counterclockwise around its furthermost point on the right shown in the picture and then translated 4 units to the right.

500

1. Draw as many different triangles as you can with each of these sets of conditions:

      a.  One angle measures 40 degrees, one side  measures 4 cm, and one side measures 5 cm.

      b.  Two sides measure 6 cm, and one angle measures 100 degrees.

2.  Did either of these sets of measurements determine one unique triangle? How do you know?

1. 

a. There are 4 different triangles that can be drawn from the given conditions.


b. There is only one triangle that can be drawn from the given conditions.

2. The second set of measurements determined one unique triangle. There was no other way I could draw it.

500

At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from the slide.

  1. Draw a diagram of the situation including a place where Lin could be.
  2. How far away from the swings is Lin in your diagram?
  3. Where are some other places Lin could be?

2. There is no way to know for sure, because we don’t know what direction Lin is from the slide. She could be anywhere between 2 and 8 meters away from the swings.

3. Lin could be at any position along a circle that is centered on the slide and that has a radius of 3 meters.