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Rigid Motions Proof – Cheat Sheet
Key Concepts
Refresher-
Additional Notes
List all the Rigid Motions we’ve learned:
Mapping Figures
The idea of pairing each vertex of the preimage to its corresponding point on the image.
How we use mapping to prove congruence: If we can map every single point of the preimage onto its corresponding points on the image (w/ rigid motions), then the figures are
congruent.
Proof TYPE 1: Describing the rigid motions that will result in congruent figures:
Example: Describe what type of rigid motion(s) you would use to map ΔABC onto ΔDEF.
Important! YOU MUST INCLUDE
THIS WHEN DESCRBING A:
1. Reflection -> line/segment of
reflection.
2. Rotation-> Around what point?
Orientation? Where do you stop
rotating? So that __ goes to __.
3. Translation -> along VECTOR!
First, translate Triangle ABC along vector AD, such that A maps to D. Then, Rotate Triangle
A’B’C’ around point A’ such that AC maps onto DF. Lastly, Reflect Triangle A’’B’’C’’ over
A’’C’’, such that B’’ maps onto E’’.
Are the two triangles congruent? Explain your reasoning.
Yes, these triangles are congruence since there is a sequence of rigid motions that allow all
corresponding sides to be mapped onto one another.
Proof TYPE 2: Two Column Rigid Motion Proof
How do we recognize them? Look for trigger words!
IMAGE LINE OF REFLECTION
REFLECTED OVER TRANSFORMATIONS
RIGID MOTIONS
Fact Check: In an isosceles triangle, if a perpendicular bisector is drawn in, it is
also the ______________________________ of the vertex angle.
So, If we are given : 1. BD is Angle bisector in an isosceles triangle
Then we know:2. _________________________________
Which means we also have a 3. _________________________________
Example: Seen below!
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Regents-Style Questions
1. Given: Isosceles DABC , BA @ BC and BD , the angle bisector of ÐB
Prove: DABD @ DCBD
Statements
Reasons
1. Isosceles DABC , BA @ BC and
BD , the angle bisector of ÐB
2. BD is the perpendicular bisector
1. Given
3. BD is the line of reflections
3. Perpendicular bisectors are lines of reflection
4. AD ≅ CD
4. Definition of perpendicular bisector
5. A maps onto C
5. AD ≅ CD and bd is the line of reflection
6. B maps onto B
6. B is on the lines of reflection
7. D maps onto D
7. D is on the lines of reflection
8.
DABD @ DCBD
Try Another! In the diagram of
a) Prove that
2. In Isosceles Triangles, the perpendicular
bisectors and angle bisectors coincide.
8. All corresponding vertices of the preimage
and image can me mapped onto each other
through rigid motions.
and
below,
,
, and
. Why is this a shortcut proof?
b) Describe a sequence of rigid motions that will map
onto
.
.
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Right Triangle Trig – Cheat Sheet
Key Ideas
Radicals:
-Rationalize Denominators
2
2√7
√7
Simplify
⋆ =
7
√7 √7
Trigonometric Ratios – SOHCAHTOA ****PUT CALCULATOR IN DEGREES***
 Used to solve for sides
or to solve for angles
 When solving for an angle
remember to use: sine,
cosine, or tangent
*Angle of Elevation – angle
formed from ground up
*Angle of Depression – angle formed from eye-level down
Auxiliary lines to solve in non-right triangles
Tips:
 Read carefully, if it is NOT a right triangle we can’t use SOHCAHTOA!
 Auxiliary lines will always be an ALTITUDE to help us get right triangles so
we CAN use SOHCAHTOA
 *NOTE: An altitude is not always an angle bisector
Try it! See the sketched auxiliary line to help create right triangles in the diagram:
Cofunctions
sin 𝜃 = cos(90 − 𝜃)
cos 𝜃 = 𝑠𝑖𝑛(90 − 𝜃)
Try it! Write a trig function equivalent to the following, but with an angle value less
than 45°
sin 78 = cos(90 − 78) = cos (12)
Additional Notes
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Determine Exact values Trig Values in Special Right Triangles
PRACTICE !
45-45-90
Try it!
30-60-90
(cos 60°)(sin 45°)
Trigonometric Identities
tan 𝜃 =
sin 𝜃
cos 𝜃
Try it! Simplify the following trig expressions:
a)
tan 
sin
=
sin 𝜃
cos 𝜃
sin 𝜃
sin 𝜃
1
1
= cos 𝜃 ∙ 𝑠𝑖𝑛𝜃=𝑐𝑜𝑠𝜃
Regents-Style Questions
1. In
, the complement of
1)
2)
3)
4)
is
. Which statement is always true?
2. Take a look at a Student Sample…
**It pays to show your work, even the set-ups!!
Let’s Try One!!
A flagpole casts a shadow 16.60 meters long. Tim stands at a distance of 12.45 meters from the base of
the flagpole, such that the end of Tim's shadow meets the end of the flagpole's shadow. If Tim is 1.65
meters tall, determine and state the height of the flagpole to the nearest tenth of a meter.
Highlight important details as we go through this!!
Area on the Plane - Cheat Sheet
Key Topics
Additional Notes
Graphing inequalities:
Describing regions using inequalities:
Be careful with vertical and horizontal lines!
Watch out with the direction of shading!
Using inequalities, describe the polygonal region
below:
Area of Overlapping polygons
Area of Overlapping Regions
Remember!
Union (∪)- Add area of each shape minus overlap
Intersection(∩)- Area of JUST the overlap
Area of Slanted Polygons
shown
 Draw a “box” around slanted polygon.
 Label each region with Roman numeral (including the missing region).
 Calculate the area of each region and subtract form the area of the “box”
Approximating the area of curved Regions:
Approximate area of curved Regions
Remember! Under Approx (stay UNDER the line or inside shape)
Over Approx ( stay over the line or outside of shape)
Take the
Average!
Remember: The SMALLER the boxes we use inside the curved region, the BETTER the approximation!
Example:
Finding the shortest distance between a point and a line.
Memorize!
Review Session 3 Practice Problems
1. As graphed on the set of axes below,
is the image of
sequence of transformations.
Is
congruent to
your answer.
after a
? Use the properties of rigid motion to explain
2. Describe a sequence of rigid motions that will map ∆𝐴𝐵𝐶 𝑡𝑜 ∆𝐷𝐸𝐹
3. Given: G is the image of E after a reflection over DF and ∆𝐷𝐸𝐹 𝑎𝑛𝑑 ∆𝐷𝐺𝐹 are drawn
Prove DEF  DGF using rigid motions
Statements
Reasons
1.
1. Given
2.
2.
3.
3.
4.
4.
4. a) List a sequence of rigid motions that will map ∆𝐽𝐾𝐻 𝑡𝑜 ∆𝐹𝐺𝐷 . (You don’t have to describe here)
b) Use the properties of rigid motions to explain why ∆𝐽𝐾𝐻 ≅ ∆𝐹𝐺𝐷
5.
Use the diagram shown where ∆LAC≅ ∆DNC for parts A and B.
a)Fill in:
Mapping:
The rigid motion RC D maps onto __
The rigid motion RC N maps onto __
The rigid motion RC C maps onto __
6. In the diagram of DLAC and DDNC below, LA @ DN , CA @ CN , and DAC ^ LCN .
. Describe a rigid motion(s) that will map DLAC onto DDNC .
7. Given: C is the image of A after a reflection over BD ∆𝐴𝐵𝐷 𝑎𝑛𝑑 ∆𝐶𝐵𝐷 are drawn.
Prove using rigid motions: ABD  CBD
Complete the proof:
ABD  CBD
Statements
Reasons
1. : C is the image of A after a
reflection over BD and
and CBD are drawn.
ABD
1. Given
2.
2.
3.
3.
4.
ABD  CBD
4.
8.
In the diagram below,
and points A, C, D, and F are collinear on line .
Let
(not drawn) be the image of
mapped onto point A.
after a translation along , such that point D is
What will the next rigid motion be in this sequence to map ∆D’E’F’ onto ∆ABC?
Are these triangles congruent? Why or why not?
9. In scalene triangle ABC shown in the diagram below,
Which equation is always true?
1)
2)
3)
4)
Sin A = Sin B
Cos A = Cos B
Cos A = Sin C
Sin A = Cos B
.
10.
11.
In a right triangle ABC, with right angle C, Sin A = Cos 9A. Using this, A is equal to…
If Sin(x+20) = Cos(x), find the value of x.
12. In the diagram of right triangle ABC shown below,
What is the measure of
and
.
, to the nearest degree?
13. A carpenter leans an extension ladder against a house to reach the bottom of a window 30 feet above the
ground. As shown in the diagram below, the ladder makes a 70° angle with the ground. To the nearest
foot, determine and state the length of the ladder.
14. As shown in the diagram, the angle of elevation from a point on the ground to the top
of the tree is 34°. If the point is 20 feet from the base of the tree, what is the height of
the tree, to the nearest tenth of a foot?
15. This is a toughy – but was a regents question!! Cathy wants to determine the height of the flagpole shown
in the diagram below. She uses a survey instrument to measure the angle of elevation to the top of the
flagpole, and determines it to be 34.9°. She walks 8 meters closer and determines the new measure of the
angle of elevation to be 52.8°. At each measurement, the survey instrument is 1.7 meters above the
ground.
Determine and state, to the nearest tenth of a meter, the height of the flagpole.
16. The diagram below shows a ramp connecting the ground to a loading platform 4.5 feet above the ground.
The ramp measures 11.75 feet from the ground to the top of the loading platform.
Determine and state, to the nearest degree, the angle of elevation formed by the ramp and the ground.
17. As shown in the diagram below, a ship is heading directly toward a lighthouse whose beacon is 125 feet
above sea level. At the first sighting, point A, the angle of elevation from the ship to the light was 7°. A
short time later, at point D, the angle of elevation was 16°.
To the nearest foot, determine and state how far the ship traveled from point A to point D.
18. In the accompanying diagram of right triangles ABD and DBC,
length of
,
, leave answer in simplest radical form.
19. Rationalize the following:
A)
14
√7
B)
√8
√2
20. Which ratio represents the cosine of angle A in the right triangle below?
a)
3
5
b)
5
3
c)
4
5
d)
4
3
, and
. Find the
Know the difference! Solving for an angle and solving for a side!!
21.
a) A tree casts a 25-foot shadow on a sunny day, as
shown in the diagram below
b) The diagram below shows the path a bird
flies from the top of a 9.5 foot tall sunflower to a
point on the ground 5 feet from the base of the
sunflower.
If the angle of elevation from the tip of the shadow to the
top of the tree is 32°, what is the height of the tree to the
nearest tenth of a foot?
To the nearest tenth of a degree what is the
measure of angle x?
21.
Find the exact value of:
A) (sin45)(cos45)(tan60)
B)
𝑐𝑜𝑠30
𝑠𝑖𝑛45
22. Simplify each expression:
2
a) 1 cos 
b ) tan cos
tan  cos 
sin
c)
23 .Given triangle 𝐷𝐸𝐹, ∠𝐷 = 22°, ∠𝐹 = 91°, 𝐷𝐹 = 16.55, and 𝐸𝐹 = 6.74,
a) find 𝐷𝐸 to the nearest hundredth.
b) Solve for the area of triangle DEF. (Hint: You need the altitude).
24. Graph the solution set for the inequality
is in the solution set. Justify your answer.
on the set of axes below. Determine if the point
25. Using inequalities, describe the polygonal region shown below:
26. Using inequalities, describe the region with a system of inequalities. (Note: here it is not an enclosed
figure)!
27.
a) Determine the region that satisfies the following system of linear inequalities:
y≤x
y≥2
x<5
b) What polygonal region is formed by the solution set?
c) What is the area of that polygon?
28. Determine the approximate area of the following curved region.
29. You are asked to approximate the area of the US. After copying a replica of the outline of the US using a
scale, you realize in your desk you only have two types of graph paper to use! Which piece of graph paper
would get a better estimated value for the area of the US? Justify your answer!
30. In the accompanying diagram, right triangle ABC is inscribed in a circle. BA is the diameter, BC =6
cm, AC = 8 cm. Find the exact area of the shaded region. Leave your answer in terms of 𝜋
31. Polygon A is a square and Figure B is a circle. Calculate the area of A U B. Round your answer to the
nearest inch.
32. Determine the area of the following quadrilateral
33. Determine the area of a triangle ABC, whose vertices are A(-3,1) , B(1,3) and C(3,-2).
34.
Determine the shortest distance between the line and the given point, leave answer in radical form.
1
35. Determine the distance between a point and a line:𝑦 = − 2 𝑥 + 2 and point (4,-2) round to the
nearest whole number.
Hi Geometry Friend! You worked hard!
Check your work to see if you did it
right!