Math 391: Geometric Reasoning
Which ordered pair is between the x-axis and the line j? (2, 8) (2, 3) (-2, -1) (-2, 5)
(2, 3) Line j is a horizontal line through the point (0,5) on the y-axis. This is the graph of the equation y = 5. Only point (2,3), in quadrant I, is between the x-axis and line j.
What is the measure of angle y? 66° 114° x° 180°
114° Angle Y because it is supplementary with known angle 66°. Therefore, 180°-66° = y = 114°.
The images below are similar triangles. Solve for the missing value, x. 15 11 20 12
12 Since the triangles are similar, this question is best solved by setting up a proportion: 15/20=x/16 Solving leads to 20x=240. This reduces to x=12.
What is the measure of angle x? 114° 66° 180° y°
66° Angle x is a vertical angle with known angle 66° so they must be equal.
Two angles are complementary. If the measure of one of the angles is 68°, what is the measure of the other angle? 68° because complementary angles are congruent to each other. 112° because the sum of the measures of complementary angles is 180°. 292° because the sum of the measures of complementary angles is 360°. 22° because the sum of the measures of complementary angles is 90°.
22° because the sum of the measures of complementary angles is 90°. Complementary angles are two angles whose sum is 90 degrees. So, if one angle is 68 degrees, its complement would be 90 - 68 = 22 degrees.
If the measure of angle 1 = 70°, which angle also measures 70°? 2 3 4 None of the above
3 Angles 1 and 3 are vertical angles. Vertical angles are made with 2 intersecting lines that cause two opposite pairs of congruent angles. Congruent means equal, therefore, making angle 3 also measure 70°.
Marvin is trying to get to his friend's house. He walks 4 blocks north to the park, then turns right and walks 3 blocks. Finally he turns right and walks 4 blocks. How far from his starting point does he wind up? 11 blocks northeast 3 blocks west 3 blocks east 11 blocks northwest
3 blocks east Turning right while walking north makes Marvin travel east. He travels east by 3 blocks. When he takes another right he heads south. He winds up in the same north/south position, but 3 blocks east.
Mrs. Perkins is beginning to teach her class about congruent shapes. Which of the activities below is the best activity to introduce the subject? Give students a small quiz to access their current knowledge about congruent shapes. Give students geometric proofs that have been scrambled so the steps are out of order and have them reorder them. Have students copy notes with important key terms related to congruent shapes. Allow students to use cutouts of shapes that have been magnified to different dilations and compare and contrast their attributes.
Allow students to use cutouts of shapes that have been magnified to different dilations and compare and contrast their attributes. This teaches students at the concrete level since they are interacting with physical manipulatives. This is the best first activity.
Mr. Marlowe wants his students to be able to interpret word problems relating to geometry. He has many ELL students. What is the best way to clarify the meaning of various prepositions such as: above, below, together, apart, inside, etc.? Teach students to underline the confusing words in a question and ask native speakers for clarification. Give students the questions in their native language. Give students a dictionary to use while solving word problems. Give students a visual diagram that explains these terms.
Give students a visual diagram that explains these terms. Visual diagrams will help students understand these terms.
Mrs. Blue wants her students to be able to write two column geometric proofs. Which is the most appropriate way to determine their mastery? Have students write an essay about writing proofs. Give students an open ended exam where they write multiple two column proofs. Give students a proof with steps missing and ask them to write in the missing reasons. Give students a multiple choice exam related to proofs.
Give students an open ended exam where they write multiple two column proofs. Asking students to write a proof is the best way to determine if they can write a proof.
Ed graphs the point (3, 4) on the coordinate plane. Which quadrant is the point in? I II III IV
I Since both coordinate values are positive, (3,4) is in the first quadrant.
In which quadrant is the point (-5, -3) located? I II III IV
III Points with negative x and y coordinates are located in quadrant III.
Eddie graphs the point (-4, -8) on the coordinate plane. Which quadrant is the point in? I II III IV
III Since both coordinate values are negative, (-4, -8) is in the third quadrant.
Which of the following explains why the Pythagorean Theorem is a theorem? It is a definition. You can draw a picture. It is self-evident. It required a proof to be accepted.
It required a proof to be accepted. All theorems must be proven using axioms, postulates, and other theorems to be accepted.
Based on the figure, which of the following is true? ∠AGE and ∠CHE are supplementary. Line EF is the transversal of lines AB and CD. ∠AGE is complementary to ∠CHF. ∠AGE and ∠CHE are alternate interior angles.
Line EF is the transversal of lines AB and CD. Because ∠AGE and ∠CHE are congruent, they are corresponding angles. Therefore, it can be inferred that lines AB and CD are parallel and line EF is the transversal.
Ms. Todd gives students a project where she gives all students in her class a single set of ordered pairs numbered 1 through 30. They need to graph ordered pairs in order and then connect the dots in the order in which they are graphed to make a picture. This serves as their final unit project on graphing points on the coordinate plane. Is this a suitable project? No, because students can easily copy each others work. Yes, because it allows students to demonstrate mastery. No, it doesn't demonstrate mastery of the concept. Yes, because it is high interest to students.
No, because students can easily copy each others work. Since all students receive the same ordered pairs copying may be rampant and students may not be demonstrating mastery.
Mr. Macrow's first-grade class is having a hard time understanding prepositions for directionality. What is the most effective lesson for his students? Have each student draw one term and create a gallery wall. Provide an anchor chart and objects. Have students move two objects to form each relationship on the chart. Create flashcards of the terms for students to practice. Draw images of each phrase while discussing them.
Provide an anchor chart and objects. Have students move two objects to form each relationship on the chart. Young children learn a new concept best using a concrete manipulative. By providing objects to move and describe, students can connect the vocabulary to the directionality of the two objects.
There is a line AB defined by two points A and B. If Carly draws a point M that is not on line AB, which geometric figure is defined by the point M and the line? a vertex a plane an angle two lines
a plane A plane is defined by non-collinear points. In this case points A, B, and M do not reside on the same line; therefore, they define a plane.
In the figure, which pair of line segments is most likely perpendicular? segment AB and Segment DC segment AB and segment BC segment AE and segment AB segment CD and segment CE
segment AB and segment BC Segment AB and segment BC appear to be perpendicular; they seem to intersect at a right angle to each other.
For △ABC, it is given that m∠A=50°° and m∠B=40°. It can be determined that m∠C=90° because of which of the following? A the converse of the Pythagorean Theorem B the postulate that states "The sum of the interior angles of a triangle is 180º." C the Pythagorean Theorem D the theorem that states, "The sum of the interior angles of a triangle is 180º."
the theorem that states, "The sum of the interior angles of a triangle is 180º." "The sum of the interior angles of a triangle is 180º" is a theorem because it required a proof to be accepted. It is needed to create and solve the equation 50°+40°+m∠C=180°.
Find the line parallel to the line x=-4 that goes through (2, 6). x=6 x=2 y=-4x+14 y=6
x=2 Since x=-4, this is a vertical line and there is no slope. The line parallel to it will also be vertical so, if it goes through (2, 6), the equation of the vertical line is x=2.
A line passes through the point (-1, 2) and is parallel to the line 2x + y=6. What is the equation of the line? y = 2x - 4 y = 2x +2 y = -2x + 4 y = -2x
y = -2x First, solve the given equation for y: y = -2x +6. So the slope is -2 and the line parallel to this line also has a slope of -2. Use the point-slope formula: y−y^1=m(x−x^1). Substitute m = -2 and the point (-1, 2) for (x^1,y^1). The equation becomes y−2=−2(x−(−1)). Simplify and the equation is y−2=−2(x+1), then y−2=−2x−2, which then becomes y=−2x.
Find the equation of the line perpendicular to y = - ¼ x +2 that goes through the point (3, -1). y=-4x+11 y=4x-11 y= -¼ x - ¼ y=4x -13
y=4x -13 The given equation has a slope of m= -¼, so the slope of the line perpendicular to it will have a slope of 4. Perpendicular lines have slopes that are opposite reciprocals of each other. Substitute the point and �=4m=4 into the point-slope formula, y−y1=m(x−x1) : y−(−1)=4(x−3). This becomes y+1=4x−12. Then subtract 1 from both sides to solve for y: y=4x-13.
Find the line that is parallel to y-5x= 7 and passes through (-2, 3) y=−1/5x-2/5 y=5x+13 y=5x+7 y= -5x +7
y=5x+13 First, solve the equation for y by adding 5x to both sides so, y=5x+7. Parallel lines have the same slopes so use the slope m=5 and the point and substitute into the point-slope formula: y−y^1=m(x−x^1). So, y−3=5(x−(−2)) becomes y−3=5(x+2). Then after distribution, y−3=5x+10. Solve for y by adding 3 to each side: y=5x+13.
A line passes through the point (4, 2) and is perpendicular to the line 3x-y=6. What is the equation of the line? y=−1/3x+10/3 y=1/3x+4/3 y = -3x + 14 y=3x-10
y=−1/3x+10/3 First, solve the equation for y to find the slope: y = 3x - 6. The slope of the line is 3 so the slope of the perpendicular line is m=- ⅓ . Perpendicular lines have slopes that are opposite reciprocals of each other. Use the point-slope formula: y−y1=m(x−x1). Substitute m =- ⅓ and the point (4,2) so that the formula becomes: y−2=−1/3(x−4)→y−2=−1/3x+4/3→y−6/3+6/3=−1/3x+4/3+6/3 When simplified the equation of the line is y=−1/3x+10/3.
Based on the image below, if triangle DEF is congruent to triangle GHI, which of the statements is true? ∠D ≌ ∠I ∠ E and ∠F are complementary Side DE ≌ Side EF ∠G and ∠F are complementary
∠G and ∠F are complementary If two triangles are congruent, then corresponding sides (sides that are in the same position) and corresponding angles (angles that are in the same position) are also congruent. In the figure, this means that DE = GH, DF = GI, and EF = IH and m∠D ≌ m∠G, m∠F ≌ m∠I, m∠E ≌ m∠H. The symbol in the corner of ∠E and ∠H tells us that E and H are right angles and are each equal to 90°. Since the sum of the angles in any triangle is 180°, this means that since m∠H = 90°, then m∠G + m∠I = 180 - 90 = 90° and would, therefore, be complementary. But what about ∠G and ∠F? Well, since ∠F and ∠I are corresponding angles, they are congruent and their measures can be substituted for each other. So, m∠G + m∠F = 90°. Therefore, this option is a true statement.
