Semester Test Topic Review. Correct Version

Transcription

1 Semester Test Topic Review Correct Version

2 List of Questions Questions to answer: What does the perpendicular bisector theorem say? What is true about the slopes of parallel lines? What is true about the slopes of perpendicular lines? What is slope intercept form of an equation of a line? What do m and b represent? What is point slope form of the equation of a line? What do x1, y1, and m represent? What does it mean if two lines intersect on the x-axis? What does it mean if two lines are coinciding? What does a translation by the vector <3, -5> do to an image? What about <-2, 4>? What happens to each ordered pair (x, y) if it is rotated 90 degrees counterclockwise? What six things do you know if you are given a triangle congruence statement? What does the third angles theorem say? What is true about the non-right angles of a right triangle? How many obtuse angles can one triangle have? How many right angles can one triangle have? What are base angles of an isosceles triangle? What is the vertex angle? What are the legs of an isosceles triangle?

3 List of Vocabulary words Definitions: acute angle adjacent angles perpendicular bisector alternate exterior angles alternate interior angles consecutive interior angles corresponding angles vertical angles reflection rotation translation inductive reasoning isosceles triangle SAS SSS ASA AAS HL midpoint segment bisector angle bisector equilateral triangles incenter orthocenter incenter centroid altitude median base angles of isosceles triangles legs of isosceles triangles

4 Midsegment of a triangle We have not gone over this in class, but it is on your semester test. Because J is the midpoint of FH and K is the midpoint of HG, JK is a midsegment of triangle HGF. The following things are then true: FJ and JH are congruent and HK and KG are congruent. FG is two times the length of JK. JK is half the length of FG. FG and JK are parallel.

5 What does the perpendicular bisector theorem say? If a point is on the perpendicular bisector of a segment (like C is in the picture), then that point is equidistant from the endpoints of the segment.

6 What is true about the slopes of parallel lines? Parallel lines are coplanar lines that never intersect. They have the same slope.

7 What is true about the slopes of perpendicular lines? Perpendicular lines intersect to form right angles. Slopes of perpendicular lines are opposite (one is negative and one is positive)and reciprocals (½ and 2).

8 What is slope intercept form of an equation of a line? What do m and b represent? Slope intercept form is y=mx+b. The y-intercept, b, is the point where the line touches the y-axis. The slope, m, is the steepness of the line.

9 What is point slope form of the equation of a line? What do x1, y1, and m represent? Point-slope form of a line is y-y1=m(x-x1) and is used when you are given a point and the slope or given two points. The slope, m, is the steepness of the line, and (x1, y1) represents any point on the line.

10 What does it mean if two lines intersect on the x-axis? When two lines intersect on the x-axis, they have the same x-intercept. To find the x-intercept of a line, set y = 0 and solve for x.

11 What does it mean if two lines are coinciding? Lines that are coinciding have the same simplified equation. They are the same line. 2y=4x+6 and y=2x+3 are examples of coinciding lines.

12 What does a translation by the vector <3, -5> do to an image? What about <-2, 4>? The first number in the translation vector moves it left(-) or right(+) and the second number moves it up (+) or down(-). <3, -5> means the figure moves right three units and down 5 units and <-2,4> means the figure moves left 2 units and up 4 units.

13 What happens to each ordered pair (x, y) if it is rotated 90 degrees counterclockwise? A 90 degree counterclockwise rotation switches x and y and changes the sign of the original x. So each point (x, y) becomes (-y, x).

14 What six things do you know if you are given a triangle congruence statement?

15 What does the third angles theorem say? The third angles theorem states that if two angles of one triangle are congruent to two angles of a second triangle, then the third angles are also congruent. In the picture, angle C is congruent to angle J by the third angles theorem.

16 What is true about the non-right angles of a right triangle? The non-right angles of a right triangle are always acute angles. They are also always complementary to each other (their sums are 90 degrees).

17 How many obtuse angles can one triangle have? A triangle can have at most one obtuse angle. If there were two angles that measured more than 90 degrees, their sum would be more than 180 degrees and the triangle-angle sum theorem would be violated.

18 How many right angles can one triangle have? A triangle can have at most one right angle. If there were two angles that measured 90 degrees, their sum would be 180 degrees. Triangles have to have a third angle, but adding another angle would violate the triangle-angle sum theorem.

19 What are base angles of an isosceles triangle? What is the vertex angle? In an isosceles triangle, the base angles are the congruent angles that are across from the legs and the vertex angle is the one that is across from the non congruent side.

20 What are the legs of an isosceles triangle? The legs of an isosceles triangle are the congruent sides. They are opposite the base angles.

21 Acute Angle An acute angle is an angle whose measure is less than 90 degrees and more than 0 degrees.

22 Adjacent Angles Adjacent angles are angles who share a common side, but no common interior points (they do not overlap).

23 Perpendicular Bisector A perpendicular bisector is a segment, line, or ray that passes through a segment at its midpoint and is perpendicular to it. In the picture, segment AB is a perpendicular bisector of segment PQ because it divides PQ in half and is perpendicular to PQ.

24 Alternate Exterior Angles Alternate exterior angles are formed when two lines are crossed by a transversal. They are on different sides of the transversal and are both exterior. If the lines crossed are parallel, then the alternate exterior angles are congruent.

25 Alternate Interior Angles Alternate interior angles are formed when two lines are crossed by a transversal. They are on different sides of the transversal and are both interior. If the lines crossed are parallel, then the alternate interior angles are congruent.

26 Consecutive Interior Angles Consecutive interior angles are formed when two lines are crossed by a transversal. They are on the same side of the transversal and are both interior. If the lines crossed are parallel, then the consecutive interior angles are supplementary.

27 Corresponding Angles Corresponding angles are formed when two lines are crossed by a transversal. They are on the same side of the transversal. One is interior and one is exterior. If the lines crossed are parallel, then the corresponding angles are congruent.

28 vertical angles Vertical angles are nonadjacent angles formed when two lines intersect. Vertical angles are always congruent. In the picture the red angles are vertical and the blue angles are vertical.

29 Reflection In geometry, a reflection is a mirror image of a shape or figure. y = # (horizontal line) x = # (vertical line) y = x (exchange x and y) y = -x (exchange x and y and change the signs of both)

30 Rotation A rotation is a turn. Counterclockwise rotations are positive. Clockwise rotations are negative.

31 Translation A translation is a slide left or right and/or up or down.

32 inductive reasoning Inductive reasoning recognizes a pattern given specific instances and then generalizes this pattern to come up with a rule or to predict the next item in the sequence.

33 Isosceles triangle An isosceles triangle has at least two congruent sides. We usually use this definition to mean that it has exactly two congruent sides, but,equilateral triangles are, by definition, isosceles and have all the properties that isosceles triangles have.

34 SAS - Side angle Side Used to prove triangle congruence when two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle.

35 SSS - Side Side Side Used to prove triangle congruence when three sides of one triangle are congruent to three sides of a second triangle.

36 ASA - Angle Side Angle Used to prove triangle congruence when two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle.

37 AAS - Angle Angle Side Used to prove triangle congruence when two angles and the non-included side of one triangle are congruent to two angles and the non-included side of a second triangle.

38 HL - Hypotenuse Leg Used to prove triangle congruence in RIGHT TRIANGLES when the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and leg of a second right triangle.

39 midpoint The point halfway between the endpoints of a segment. The midpoint divides a segment into two congruent segments. B is the midpoint of segment AC in the picture and divides segment AC into two congruent segments, AB and BC.

40 Segment Bisector A segment, line, or ray that intersects a segment at its midpoint. Line n is a segment bisector of segment PR.

41 Angle Bisector An angle bisector divides an angle into two congruent angles. In the picture ray OY is an angle bisector of angle XOZ because it divides it into two congruent angles, XOY and YOZ.

42 Equilateral Triangle A triangle with three congruent sides. Equilateral triangles are always equiangular. So, equilateral triangles always have three 60 degree angles.

43 Incenter Incenter is the point where the angle bisectors of a triangle meet. In the picture Point K is the incenter of triangle NMO.

44 Orthocenter The point where the altitudes of a triangle meet is called the orthocenter.

45 Circumcenter The point where the perpendicular bisectors of a triangle meet is called the circumcenter.

46 Centroid The point where the medians of a triangle meet is called the centroid.

47 Altitude The altitude of a triangle starts at a vertex and is perpendicular to the line containing the opposite side. Segment AD is an altitude of triangle ABC.

48 Median A segment of a triangle that starts at a vertex and ends at the midpoint of the opposite side.

49 Base angles of isosceles triangles The congruent angles in an isosceles triangle. They are always opposite the legs.

50 legs of isosceles triangles The congruent sides of an isosceles triangle. They are always opposite the base angles.