Word Bank Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Included (used multiple times) Preserve distance and angle measures Preserve angle measures Congruent (used multiple times) Reflexive Property Before After Side-Side-Side (SSS) Angle-Side-Angle (ASA) Side-Angle-Side (SAS) Angle-Angle-Side (AAS) Hypotenuse-Leg (HL) Rigid Motions Transformations Dilations

Select words from the word bank above to correctly fill out each statement below. Not all words are used.

1. After two triangles are proven congruent, you can use CPCTC to prove pairs of sides and angle are congruent.

2. If two pairs of sides and the included angle of two triangles are congruent, then two triangles are 211. This is the SAS Postulate.

Draw and label a diagram of two triangles that shows the SAS Postulate:

3. If two pairs of angles and the included side of two triangles are congruent, then the two triangles are congruent. This is the ASA Postulate.

Draw and label a diagram of two triangles that shows the ASA Postulate:

4. If three pairs of sides of two triangles are congruent, then the two triangles are congruent. This is the SSS Postulate.

Draw and label a diagram of two triangles that shows the SSS Postulate:

$\triangle ABC \cong \triangle DEF$

5. If the hypotenuse and one leg of a pair of right triangles are congruent, then the two triangles are congruent. This the HL Theorem.

Draw and label a diagram of two triangles that shows the HL Theorem:

$\triangle ABC \cong \triangle A'B'C'$

6. If two pairs of angles and a non-included side of two triangles are $\cong$, then the two triangles are congruent. The is the AAS Theorem.

Draw and label a diagram of two triangles that shows the AAS Theorem:

$\triangle STU \cong \triangle PQR$

7. Rigid Motions create congruent figures because they preserve distance & measure

8. Dilations create similar figures because they preserve $\neq$ measures

9. Justification for Segment Bisector: A Segment Bisector divides a line segment into two congruent parts.

10. Justification for Midpoint: A midpoint divides a line segment into two congruent parts.

Unit 4 Regents Vocabulary Review- Congruent Triangles

11. Justification for Angle Bisector: An angle bisector divides an angle into two congruent parts.

12. Right angle justification: Perpendicular lines form right angles.

13. "All right angles are congruent."

14. Right triangle justification: "A triangle with one right $\angle$ is a right triangle."

15. If two angles or segments are literally the same, those angles or segments are congruent by the Reflexive Property.

16. A rotation of any multiple of $\frac{360}{n}$ degrees around the center of a regular polygon will map the polygon back onto itself. (n = number of sides)

17. Draw and label a diagram that represents reflexive property with angles. Mark the angle that is congruent because of the reflexive property.

$$\angle B \cong \angle B \ \text{by Reflexive Property}$$

18. Draw and label a diagram that shows angles that could be marked congruent because of vertical angles. Mark each pair of angles congruent.

$$m \angle ABC \cong \angle DCE \ \text{Vertical Angles} \ \text{arc RS}$$

19. Draw and label a diagram that represents reflexive property with segments. Mark the segment that is congruent because of the reflexive property.

$$\overline{BD} \cong \overline{BD} \ \text{by Reflexive Property}$$

Other Reasons to Know:

20. "Congruent segments added to congruent segments are congruent."

21. "Congruent segments subtracted from congruent segments are congruent."

22. "Segments that are half the length of congruent segments are congruent."

23. "Segments that are twice the length of congruent segments are congruent."

Matching Word Bank – Note: you may not use every term

dilations translations A median A reflection the same as regular different than rectangle rhombus square parallel 60 degrees 120 degrees 72 degrees standard perpendicular equal negative reciprocal point-slope same-side exterior same-side interior half midsegment vertical angles centroid isosceles equilateral largest smallest Rigid Motions Dilations (3, -7) (-3, 7) (5, -7) parallel a linear pair supplementary point-slope alternate exterior corresponding congruent alternate interior

1. $\triangle DBC$ and $\triangle ABC$ form a linear pair and are therefore supplementary.

2. $\triangle DBC$ and $\triangle ABE$ form vertical angles and are therefore $\cong$.

3. The centroid is the intersection point of all the triangle's medians.

4. The altitude is the perpendicular line segment drawn from the vertex to the opposite side of the triangle.

5. On a 2-dimensional plane, parallel lines will never intersect and will have $\cong$ slopes.

6. Perpendicular lines intersect to form right angles and will have negative reciprocal slopes.

7. A median is the line segment connecting a vertex to the midpoint of the opposite side of a triangle

8. Rigid motions preserve distance and angle measure.

9. Dilations always preserve angle measure. They only sometimes preserve distance. (when k=1)

10. A reflection is a transformation that can be described as a "flip". The ordering of the vertices of the image is different than the pre-image.

11. A rotation is a transformation that can be described as a "turn". The ordering of the vertices of the image is the same as the pre-image.

12. Alt int corresponding angles are always congruent when formed by two parallel lines are cut by a transversal (list all that apply)

13. Int & ext angles are always supplementary when formed by two parallel lines are cut by a transversal (list all that apply)

14. Translations are types of transformations (list all that apply) in which the image line segments will always be parallel to the preimage line segment (except in one special case) (when k=1)

15. The line segment which connects two midpoints in a triangle is called the midsegment. This segment is parallel to the third side and half its length.

16. Isosceles triangles have at least two congruent sides and two congruent angles.

17. Equilateral triangles have three congruent sides and three congruent angles.

18. The longest side of a triangle is across from the angle with the largest measure

19. The shortest side of a triangle is across from the angle with the smallest measure.

20. A regular polygon is a polygon with all sides and interior angles congruent.

21. The measure of each interior angle of a regular hexagon is 120 degrees.

22. A square is a quadrilateral that is a regular polygon

23. The equation of the line y - 7 = 5(x + 3) is written in point-slope form and the line goes through the point (-3, 7)

24. The equation of the line y = 5x + 1 is written in slope-intercept form.

25. The minimum # of degrees a reg. hexagon to rotate onto itself => $\frac{360}{6} = 60^\circ$

Word Bank

the scale factor = 1 in proportion parallel proportionally scale factor center AA~ Theorem Congruent multiplied by the scale factor 3 similar triangles "product of the means = product of the extremes" the same as the preimage corresponding point of the preimage

similar k (where k is the scale factor) $k^2$ (where k is the scale factor) congruent in proportion center Side Splitter Theorem included SAS congruent AA the same as

1. Every dilation must have a Center and a Scale factor.

2. Every point of the image of a dilation must fall along a line connecting the center to the corresponding point of the preimage.

3. Dilations always result in an image which is Similar to the preimage.

The image can also be congruent to the preimage if the scale factor = 1

4. AA~ theorem states that if two angles are congruent, then the two triangles are similar.

5. SAS~ theorem states that if two sides are in proportion and the included angle is congruent, then the two triangles are similar.

6. A dilation of a line with the center not on the line will result in a line that is parallel to the preimage line. If the center of dilation is the origin, the y-intercept will be multiplied by the scale factor

7. A dilation of a line with the center on the line will result is a line that is the same as the preimage

8. Corresponding angles of similar figures are Congruent.

9. Corresponding side lengths of similar figures are in proportion.

10. The altitude of a right triangle creates 3 similar Δ's

11. Every proof involving similar triangles on previous Geometry Common Core Regents has used AA ~ Theorem.

12. If a line intersects two sides of a triangle and is parallel to the third side of the triangle,

it divides the two sides that it intersects proportionally.

This is called the side splitter theorem Ex: $\frac{OA}{CO} = \frac{FB}{EC}$

13. This phrase can be used as the justification for cross-multiplication in a proof: The product of the means equals the product of the extremes

14. The perimeter of an image is K times the perimeter of the preimage following a dilation.

15. The area of an image is $K^2$ times the area of the preimage following a dilation.

Unit 6 Regents Vocabulary Review – Quadrilaterals

1. A Square has all the properties of a rectangle and a rhombus.

2. A rhombus is the most general description for a quadrilateral with four congruent sides.

3. A Square has four congruent sides and four congruent angles.

4. parallelogram is the most general term to describe a quadrilateral which has two pairs of opposite sides which are parallel and congruent.

5. Trapezoid is the most general term to describe a quadrilateral with only one pair of parallel sides.

6. State all quadrilaterals which have diagonals that bisect each other: Parallelogram Square, Rectangle, Rhombus (Types of parallelograms)

7. If a parallelogram has one right angle, it is a rectangle.

8. A rhombus is the most general name for a quadrilateral whose diagonals bisect opposite angles.

9. List the names of all quadrilaterals which have congruent diagonals: Rectangle Square (a type of rectangle) Isosceles Trapezoid

10. List the names of all quadrilaterals which have perpendicular diagonals: Rhombus Square (a type of rhombus) Kite

11. A square has all the properties of a rectangle and rhombus.

Converse Theorem Word Bank

Two pairs of parallel sides One pair of sides that is both parallel and congruent Two pairs of congruent opposite angles Diagonals bisect each other Four congruent sides Perpendicular diagonals Diagonals bisect opposite angles Diagonal that bisects two opposite angles One right angle Congruent diagonals Three right angles Two congruent, consecutive sides

Parallelogram Converse Theorems

15. If a quadrilateral has two pairs of parallel sides, then it is a parallelogram.

16. If a quadrilateral has one pair of sides that is both parallel and $\cong$, then it is a parallelogram.

17. If a quadrilateral has two pairs of $\cong$ opposite angles, then it is a parallelogram.

18. If a quadrilateral has diagonals that bisect each other, then it is a parallelogram.

19. If a quadrilateral has two pairs of congruent sides, then it is a parallelogram.

Rectangle Converse Theorems

16. If a parallelogram has one right angle, then it is a rectangle.

17. If a parallelogram has congruent diagonals, then it is a rectangle.

18. If a quadrilateral has three right angles, then it is a rectangle.

Rhombus Converse Theorems

19. If a parallelogram has 2 congruent, consecutive sides, then it is a rhombus.

20. If a parallelogram has a diagonal that bisects 2 opp angles, then it is a rhombus.

21. If a parallelogram has perpendicular diagonals, then it is a rhombus.

22. If a quadrilateral has 4 congruent sides, then it is a rhombus.

Word Bank (you will not use all terms and some you will use more than once) Right Triangle Isosceles Triangle the Pythagorean Theorem trigonometric ratios (Sine, cosine, tangent) Inverse trigonometric ratios (sin^{-1}x, cos^{-1}x, tan^{-1}x) Radian Degree Tangent Cosine Supplement Complement Distance Formula (equal distances) Slope Formula (negative/opposite reciprocal slopes) Midpoint Formula (the midpoint of each line segment is the same) Slope Formula (slopes are the same)

1. If you are given the measure of one side and one angle in a right triangle, you can use trigonometric ratios (Sine, cosine, tangent) to find the measure of a missing side.

2. If you are given the measure of two sides in a right triangle, you can use the Pythagorean Theorem to find the measure of a missing side.

3. If you are given the measure of at least two sides in a right triangle, you can use Inverse trigonometric ratios (sin^{-1}x, cos^{-1}x, tan^{-1}x) to find the measure of any missing angle.

4. What does SOH-CAH-TOA stand for?

$$\text{Sin}(x) = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \text{Cos}(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \text{Tan}(x) = \frac{\text{opposite}}{\text{adjacent}}$$

5. Your calculator should be set to Degree mode when you start the Regents.

6. The sine of any acute angle is equal to the cosine of its complement.

Example: sin (30) = cos (60).

7. What is the distance formula? (look up in notes)

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

8. What is the slope formula? (look up in notes)

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

9. What is the midpoint formula? (look up in notes)

$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

10. What is the formula to find the point that results from dividing a directed line segment into a given ratio? (look up in notes)

if the ratio is a:b:

$$( x_1 + \frac{a}{a+b}(x_2 - x_1), y_1 + \frac{a}{a+b}(y_2 - y_1) )$$

11. How do you prove that two line segments are congruent in a coordinate geometry proof?

Distance Formula (equal distances)

12. How do you prove that two line segments are parallel in a coordinate geometry proof?

Slope Formula (slopes are the same)

13. How do you prove that two line segments are perpendicular in a coordinate geometry proof?

Slope Formula (negative reciprocal slopes)

14. How do you prove that two segments bisect each other in a coordinate geometry proof?

Midpoint Formula (the midpoint of each line segment is the same)

Name__________________________ Unit 9 Vocabulary Review

1. Write the center-radius form of the equation of a circle:

$$r^2 = (x-h)^2 + (y-k)^2 \quad (h,k) = \text{center}$$ $$r = \text{radius}$$

Example: What is the equation of a circle with center (-1, 8) and radius 7?

$$7^2 = (x-(-1))^2 + (y-8)^2 \implies 49 = (x+1)^2 + (y-8)^2$$

2. What is the formula for the area of a sector in terms of degrees?

$$\text{Area of sector} = \frac{\theta}{360} \cdot \pi r^2$$

3. What is the formula for the area of a sector in terms of radians?

$$\text{Area of sector} = \frac{\theta}{2\pi} \cdot \pi r^2$$

4. What is the formula for arc length in terms of degrees?

$$\text{Arc length} = \frac{\theta}{360} \cdot 2\pi r$$

5. What is the formula for arc length in terms of radians?

$$\text{Arc length} = \frac{\theta}{2\pi} \cdot 2\pi r$$

6. A chord is a segment whose endpoints lie on a circle.

7. A tangent to a circle is a line that intersects the circle at exactly one point and is perpendicular to the radius at that point.

8. A secant is a line that intersects the circle at two points.

9. A central angle is the angle formed by two radii and the center of the circle. The central angle is equal to the measure of its intercepted arc.

Draw a diagram:

$$m\widehat{AB} = m\angle AOB$$

Name_________________________ Date_____ Unit 9 Vocabulary Review

10. An inscribed angle is the angle formed by two chords with a common endpoint on the circle. An inscribed angle is equal to half the measure of its intercepted arc.

Draw a diagram:

11. Arc length is measured in distance whereas arc measure is measured in degrees/radians

12. In a circle, inscribed angles that intercept the same arc (or congruent arcs) are congruent.

13. An angle inscribed in a semi-circle is a right angle.

14. Opposite angles in inscribed angles are supplementary.

15. An angle formed by the intersection of two chords is half the sum of its intercepted arcs

Name__________________________ Date_____ Unit 9 Vocabulary Review

14. An angle formed by two segments that meet outside of the circle is equal to half the difference of the intercepted arcs.

two tangents: $$m \angle B = \frac{1}{2} (240 - 120) \ = \frac{1}{2} (120) \ = 60^\circ$$

two secants: $$m \angle A = \frac{1}{2} (100 - 30) \ = \frac{1}{2} (70) \ = 35^\circ$$

a secant and a tangent $$m \angle A = \frac{1}{2} (164 - 76) \ = \frac{1}{2} (84) \ = 47^\circ$$

An angle formed by an intersecting tangent and chord is equal to half of the intercepted arc. $$m \angle ABC = \frac{1}{2} (m \overarc{AB}) \ m \angle x \ = \frac{1}{2} (148) \ = 74^\circ$$

Name__________________________ Unit 9 Vocabulary Review

15. If two chords intersect inside a circle, the product of the lengths of the segments of one chord equals the product of the segments of the other.

$$a \cdot b = c \cdot d$$

16. In a circle, parallel chords intercept $\frac{1}{2}$ arcs.

17. In a circle congruent arcs intercept congruent chords