Geometry grade 9 semester 1

Which best explains why the orthocenter of an obtuse triangle is outside the triangle?

All three of the altitudes lie entirely outside the triangle.

Angle KLM and angle MLN are a linear pair. Ray LR is to be added to the diagram so that it is opposite ray LM. Which is true about angle RLN that would be formed?

Angles RLN and MLK would be vertical angles.

Is there a series of rigid transformations that could map KLP to QNM? If so, which transformations?

Yes, KLP can be reflected across the line containing KP and then translated so that P is mapped to M.

Are the triangles congruent? Why or why not?

Yes, they are congruent by either ASA or AAS.

Is there a series of rigid transformations that could map ΔRST to ΔXYT? If so, which transformations could be used?

Yes, ΔRST can be reflected across the line containing RT and then rotated about T so that S is mapped to Y.

Which quadrilateral will always have 4-fold reflectional symmetry?

square

-Pentagon ABCDE is dilated according to the rule DO,3(x,y) to create the image pentagon A'B'C'D'E', which is shown on the graph. What are the coordinates of point A of the pre-image?

(-1, 2)

What is the pre-image of vertex A' if the image shown on the graph was created by a reflection across the line y = x?

(10, -2)

The rule R0, 90° T-1,1 (x,y) is applied to ΔBCD to produce ΔB"C"D". Point B" of the final image is at (-4, 1). What are the coordinates of point B on the pre-image?

(2, 3)

Which rule represents the translation from the pre-image, ΔABC, to the image, ΔA'B'C'?

(x, y) → (x + 7, y - 6)

If a translation of T -3, -8 (x,y) is applied to square ABCD, what is the y-coordinate of B' ?

-6

Triangle DEF is an isosceles, so DEF corresponds DFE. Angle DEF measures 75°. What is the measure of angle CFD?

105°

Tia lives at the corner of 4th Street and 8th Avenue. Lei lives at the corner of 12th Street and 20th Avenue. The fruit market is 3/4 the distance from Tia's home to Lei's home. Where is the fruit market? x = (m/m+n) (x2 - x1) + x1 y = (m/m+n) (y2 - y1) + y1

10th Street and 17th Avenue

Two parallel lines are crossed by a transversal. If m 1=61.8°, then what is the measure of m 6?

118.2°

What is QBO?

12°

What is the length of line segment LJ?

15 units

Consider the diagram. What is QS?

17 units.

A regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°. How many sides does the polygon have?

18

-Line ST and point V are shown on the graph. Line VW is to be drawn on the graph such that it is perpendicular to line ST. If the coordinates of point W are (−1, y), what is the value of y?

2

The graph shows the dilation with respect to the origin of rhombus RHOM. What is the factor of dilation?

2/3

For the triangles to be similar by the SSS similarity theorem, what must be the value of y?

20

What is the length of SA

3 ft

An isosceles right triangle has leg lengths of 4 centimeters. What is the length of the altitude drawn from the right angle to the hypotenuse?

4 cm

2-In the triangles, TR = GE and SR = FE. If GF= 3.2 ft, which is a possible measure of TS?

4.0 ft

What is the equation of the line that is perpendicular to the given line and passes through the point (3, 0)?

5x − 3y = 15

Point S lies between points R and T on RT. If RT is 10 centimeters long, what is ST?

6 centimeters

Which value of x would make NO || KJ?

8

What is the measure of JHN?

95°

Which statement describes the difference between a dilation and an isometric transformation?

A dilation changes the dimensions of a figure, while an isometric transformation preserves dimensions.

Which rule describes the composition of transformations that maps ΔABC to ΔA"B"C"?

A. R0, 90° Rx-axis (x,y)

Quadrilateral ABCD is translated up and to the right, and then rotated about point Q. Which congruency statement is correct?

ABCD ≅ ZYXW

Given that D is the midpoint of AB and B is the midpoint of AC, which statement must be true?

AC = 4DB

Which congruence theorems can be used to prove ΔEFG ≅ ΔJHG? Select two options.

ASA, AAS

Point Z is equidistant from the sides of ΔRST. Which must be true?

ASZ ≅ ZSB

Triangle JKL is transformed to create triangle J'K'L'. The angles in both triangles are shown. J = 90° J' = 90° K = 65° K' = 65° L = 25° L' = 25° Which statement is true about this transformation?

It can be a rigid or a nonrigid transformation depending on whether the corresponding side lengths have the same measures.

-Figure RHOM is a rhombus. RO and HM are the diagonals of the rhombus, as well as angle bisectors of the vertex angles, and they create four isosceles triangles: HOM, MHR, RHO, and OMR. What is true about MSR?

It must be a right angle.

Planes A and B are shown. If a new line, p, is drawn parallel to line l, which statement is true?

Line p must be drawn so that it can lie in the same plane as line l.

Triangle ABC has the angle measures shown. Which statement is true about the angles?M A=2x M B=3x M C=4x

M B=60

Based on the diagram, can point D be the centroid of triangle ACF? Explain.

No, the ratio between AD and DE is 3:1.

-Triangle RST has vertices R(2, 0), S(4, 0), and T(1, -3). The image of triangle RST after a rotation has vertices R'(0, -2), S'(0, -4), and T'(-3, -1). Which rule describes the transformation?

R0, 270°

-Four points partition the directed line segment from A to B. Point P partitions the directed line segment from A to B into a 3:4 ratio. • Point Q partitions the directed line segment from A to B into a 4:3 ratio. • Point R partitions the directed line segment from A to B into a 2:5 ratio. • Point S partitions the directed line segment from A to B into a 5:2 ratio. Which point will lie closest to B?

S

The proof that ΔABC ≅ ΔCDA is shown. Given: AB∥CD and BC∥DA Prove: ΔABC ≅ ΔCDA What is the missing reason in the proof? 1. AB ∥ CD; BC ∥ DA 1. given 2. Quadrilateral ABCD is a ▱ 2. definition of parallelogram 3. AB ≅ CD; BC ≅ DA 3. opposite sides of a parallelogram are ≅ 4. AC ≅ AC 4. reflexive property 5. ΔABC ≅ ΔCDA 5. ____________

SSS congruence theorem

If TU = 6 units, what must be true?

SU + UT = RT

A right angle intersects a line at point M. Which statement is true about angles 1 and 2?

They are complementary.

Triangle ABC was reflected over line m, then dilated by a scale factor between 0 and 1. Which diagram illustrates these transformations?

Third Image

Consider the triangle. Which statement is true about the lengths of the sides?

Two sides have the same length, which is less than the length of the third side.

Which composition of similarity transformations maps LMN to L'M'N'?

a dilation with a scale factor greater than 1 and then a translation

A line segment has endpoints at (3, 2) and (2, -3). Which reflection will produce an image with endpoints at (3, -2) and (2, 3)?

a reflection of the line segment across the x-axis

Which shows the pre-image of triangle X'Y'Z' before the figure was rotated 90° about the origin?

second image

Which is precisely defined using the undefined terms point and plane?

circle

Complete the paragraph proof. Given: M is the midpoint of PK. MB is perpendicular to PK. Prove: ΔPKB is isosceles. It is given that M is the midpoint of PK and MB is perpendicular PK. Midpoints divide a segment into two congruent segments, so PM corresponds KM. Since MB is perpendicular to PK and perpendicular lines intersect at right angles, PMB and KMB are right angles. Right angles are congruent, so PMB corresponds KMB. The triangles share side MB, and the reflexive property justifies that MB corresponds MB. Therefore, Δ PMB corresponds Δ KMB by the SAS congruence theorem. Thus, BP corresponds BK because _____________. Finally, ΔPKB is isosceles because it has two congruent sides.

corresponding parts of congruent triangles are congruent

Planes A and B intersect. Which describes the intersection of planes A and B?

line ED

-Myra took a picture of the sky one afternoon when two jet airplanes appeared to draw a pair of parallel lines with their vapor trails. The vapor trails from two other jets flying from another direction crossed over the parallel trails. She printed her picture and labeled the angles and lines. Assume lines c and d are parallel and 2 measures 98°. Which statements are true? Select three options.

m 3 = m 6 = 98°, m 4 = m 8 = 82°, m 5 = m 8 = 82°

Which set of equations is enough information to prove that lines c and d are parallel lines cut by transversal p?

m∠2 = 99° and m∠4 = 99°

What additional information could be used to prove ΔABC ≅ ΔMQR using SAS? Select two options.

m∠R = 60° and AB ≅ MQ AB = QR = 31 cm

Which figures are shown in the diagram? Select three options.

point D, ray CD, segment CD

Triangle EFG is transformed to create triangle E'F'G'. Which transformation occurred?

rotation

The triangles are congruent by the SSS congruence theorem. Which rigid transformation(s) can map FGH onto VWX?

rotation, then translation

Which distance measures 5 units?

the distance between points W and X

Given: AB = 12 AC = 6 Prove: C is the midpoint of AB. Proof: We are given that AB = 12 and AC = 6. Applying the segment addition property, we get AC + CB = AB. Applying the substitution property, we get 6 + CB = 12. The subtraction property can be used to find CB = 6. The symmetric property shows that 6 = AC. Since CB = 6 and 6 = AC, AC = CB by the _____________ property. So, AC ≅ CB by the definition of congruent segments. Finally, C is the midpoint of AB because it divides AB into two congruent segments.

transitive

Which statements about the diagram are true? Select three options.

x = 63, z = 117, x + z = 180

What is the equation of the line that is parallel to the given line and passes through the point (12, −2)?

y=5/6x-12

Triangle NLM is reflected over the line segment as shown, forming triangle ABC. Which congruency statement is correct?

ΔNLM ≅ ΔCAB

-Point E is drawn on the graph so that line EF is parallel to line CD. If the coordinates of point E are (-4, y), what is the value of y?

−8

Given: a ∥ b and ∠1 ≅ ∠3 Prove: e ∥ f We know that angle 1 is congruent to angle 3 and that line a is parallel to line b because they are given. We see that __________ by the alternate exterior angles theorem. Therefore, angle 2 is congruent to angle 3 by the transitive property. So, we can conclude that lines e and f are parallel by the converse alternate exterior angles theorem. Which information is missing in the paragraph proof?

∠2 ≅ ∠3

-In the diagram, the ratios of two pairs of corresponding sides are equal. to prove that △LMN ~ △XYZ by the SAS similarity theorem, it also needs to be shown that

∠N ≅ ∠Z

Which represents an exterior angle of triangle EGF?

∠NEG

Which piece of additional information can be used to prove that △RST ~ △VUT?

∠R ≅ ∠V