55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.


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1 Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit diagnostics are particularly helpful because they are a form of "pretest" that you can use to assess your performance and prescribe topics for review. Study the unit wrapups. Read each unit wrapup to determine which topics you need to review further. Review important vocabulary terms. Study the key terms listed on the next page to identify any vocabulary you don't know. Use the "Check Your Understanding" pages and math workbook assignments as study aids. Review the "Check Your Understanding" pages to get right, wrong, and hint feedback to specific math problems. The math workbook assignments you've completed throughout the semester will also provide you with a great review for the exam. Review the activities that contain information you need to review further. After determining which topics are least familiar to you, review the activity in which they are covered. This could include revisiting study activities and rereading math workbook assignments. Taking the Semester Exam For all exam questions Read the entire question and all the choices before selecting an answer. Then choose the best answer or answers. Semester Exam 55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points. Semester Review: Geometry, Semester 1
2 Semester Key Terms right triangle right triangle AA similarity postulate AAS theorem acute angle acute triangle altitude angle angle bisector angle bisector theorem angle measure ASA postulate base base angles bisector centroid circumcenter closed figure collinear conditional congruence transformations congruent coplanar corollary cosine cross product property deduction equilateral triangle exterior angle extremes HA theorem HL theorem hypotenuse included angle included side indirect proof induction interior angle isosceles triangle LA theorem legs length line LL theorem means median midpoint obtuse angle obtuse triangle opposite angle opposite side orthocenter parallel lines pattern perpendicular bisector theorem perpendicular lines plane point proof by contradiction proportions protractor Pythagorean theorem Pythagorean triple ratios ray remote interior angle right angle right triangle SAS postulate scale factor scalene triangle segment side similar sine skew lines SSS postulate SSS similarity theorem straight angle syllogism tangent transversal Venn diagram vertex vertical angles zero angle
3 Key Terms and Concepts for Unit 1 Unit 1 Key Terms angle conjecture contrapositive converse corollary deduction definition indirect proof induction inverse postulate proof by contradiction scientific method segment side Venn diagram vertex Know that to use induction, you look for a pattern in what you know and make a conjecture about what you don t know. Know that to use deduction, you use general rules that are certainly true and use them to find particular facts. Know that conditional statements have two parts, an "if" clause and a "then" clause. They are also called "ifthen" statements. Know that conditional statements can be illustrated using Venn diagrams and know how to interpret them. The Venn diagram below illustrates the statement "If a thing is a baseball cap, then it is a hat." Know that the converse of "If a, then b" is "If b, then a." Know that the inverse of "If a, then b" is "If not a, then not b." Know that the contrapositive of "If a, then b" is "If not b, then not a."
4 Know that if the statement "If a, then b" is true, then its contrapositive "If not b, then not a" is also true. The converse and inverse are not necessarily true. Know the shorthand for conditional statements: If a, then b If not a, then not b Know how to use syllogisms. The chain of statements below shows how to form a syllogism from two conditionals. Given the statements: If a, then b If b, then c You can conclude: If a, then c Know that a definition is the precise statement of the qualities of an idea, object, or process. Know that a postulate or axiom is a statement that is assumed to be true without proof. Know how to write a twocolumn proof. The first column contains deductive steps. The second column contains your reasons. Know how to write and understand a proof by contradiction, or an indirect proof. Start by assuming the opposite of what you want to prove and then deduce an impossible statement. Example Make a syllogism from the following statements: If there is cheese in the circus tent, then there are mice in the circus tent. If there are mice in the circus tent, then the elephants are nervous. Then write the contrapositive of the syllogism. Answer Syllogism: If there is cheese in the circus tent, then the elephants are nervous. Contrapositive: If the elephants are not nervous, then there is no cheese in the circus tent.
5 Key Terms and Concepts for Unit 2 Unit 2 Key Terms acute angle angle bisector collinear congruent coplanar length line midpoint perpendicular lines plane point protractor ray segment straight angle zero angle Know that a point has no size. A segment has no width but has finite length. A ray has no width and infinite length in one direction. A line has no width and infinite length in both directions. Know that a segment is named by its endpoints. Know that the length of a segment is written AB. Know that if the point C lies on the segment, then the sum of the lengths of and is equal to the length of. That is, AC + CB = AB. Know that a ray is named by its endpoint and another point on the ray. The endpoint comes first in the name; the other point comes second.
6 . Know that a line is named by two points on the line. or Know that the rays that form an angle are called the sides of the angle, and the point where they meet is called the vertex of the angle. Know that an angle whose sides make a line is called a straight angle. An angle whose sides overlap to make a ray is called a zero angle. Know how to use a protractor. Review Unit 2, Activity 4 for instructions and to practice using a protractor. Know that an acute angle is an angle whose measure is greater than 0 and less than 90. An obtuse angle is an angle whose measure is greater than 90 and less than 180. A right angle is an angle whose measure is exactly 90. Know that adjacent angles share a side and vertex. and are adjacent angles.
7 Know that the bisector of an angle is a ray that divides it into two angles of equal measure. is congruent to. is the bisector of. Know that a linear pair is a pair of adjacent angles that combine to form a straight angle. The angles of a linear pair are supplementary. The measures of supplementary angles add up to 180. and make a linear pair. They are supplementary. Know that complementary angles are angles whose measures add up to 90. and are complementary. Know that two segments are congruent if they have the same length. means that is congruent to. Know that two angles are congruent if they have the same measure. is equal to only if they are the same angle; is congruent to if they have the same measure. means that is congruent to.
8 Example is the bisector of. If = 24, what is? Tip It may help to sketch a diagram of the angles. Answer = 24 WorkedOut Solution Since is the bisector of, you know that. Hence = = 24. Know that if the point M is on and, then M is the midpoint of. Know that two points are collinear if they lie on the same line. Objects (including points) are coplanar if they lie in the same plane. Know that if A and B are contained in a plane, then is entirely contained in the plane. Know that vertical angles are congruent. Know that adjacent angles formed by intersecting lines make linear pairs. Know that perpendicular lines are lines that intersect at right angles. Know that the distance between a line and a point not on the line is the length of the segment that joins the point to the line and is perpendicular to the line. Know that the perpendicular bisector of a segment is the line that passes through the midpoint of the segment and is perpendicular to the segment. Know that skew lines are lines that are not coplanar. (Remember that any one line lies in an infinite number of planes; just because two lines lie in different planes doesn t mean that they are skew they are skew if there is no plane that they both lie in.) Know that parallel lines are coplanar lines that never intersect. Know that given a line and a point P not on, there is one and only one line parallel to that passes through P. Know that a transversal is a line that crosses (or "cuts") two parallel lines.
9 (Theorem) Know that for parallel lines cut by a transversal, corresponding angles are congruent. (Theorem) Know that for parallel lines cut by a transversal, alternate interior angles are congruent. (Theorem) Know that for parallel lines cut by a transversal, consecutive interior angles are supplementary.
10 Example What is the measure of? Answer WorkedOut Solution and the angle with measure 118 are consecutive interior angles, which means they are supplementary. So.
11 Key Terms and Concepts for Unit 3 Unit 3 Key Terms AA similarity postulate AAS theorem ASA postulate centroid circumcenter closed figure congruence transformations cross product property exterior angle extremes incenter included angle included side means medians opposite angle opposite side orthocenter proportion remote interior angle SAS postulate scale factor similar triangles SSS postulate SSS similarity theorem (Definition) Know that a triangle is a closed figure formed by three segments connecting three noncollinear points. The three segments are its sides, and the three points are its vertices. (Theorem) Know that the sum of the measures of a triangle s angles is always 180. If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are also congruent ( A + B + m = 180 and A + B + n = 180, so m = A  B = n ). (Definition) Know that an exterior angle is an angle that forms a linear pair with one of the interior angles of the triangle. is an exterior angle of.
12 (Definition) Know that a remote interior angle is an angle in a triangle that is not adjacent to a given exterior angle. and are the remote interior angles of. (Theorem) Know that the measure of an exterior angle of a triangle equals the sum of the measures of its two remote interior angles. The measure of an exterior angle is always greater than the measure of either of its remote interior angles. (Definition) Know that actions that you can do to a triangle that do not change its size or shape are called congruence transformations and that the three congruence transformations are translating, rotating, and reflecting. (Definition) Know that congruent triangles are triangles whose corresponding parts are congruent. (The definition of congruent triangles is often abbreviated CPCTC: "corresponding parts of congruent triangles are congruent.") Therefore: correspond.. The letters in the triangles names should be ordered to show how the vertices (Theorem) Know that congruence of triangles is reflexive, symmetric, and transitive: Reflexive property: Every triangle is congruent to itself. Symmetric property: If, then. Transitive property: If and, then.
13 (Postulate) Know that if the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. This is called the SSS postulate. (Postulate) Know that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. This is called the SAS postulate. (Postulate) Know that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. This is called the ASA postulate. (Theorem) Know that if two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of another, then the triangles are congruent. This is called the AAS theorem. Know that SSA (sidesideangle congruence) does not guarantee congruent triangles, and AAA (angleangleangle congruence) also does not guarantee congruent triangles. (Definition) Know that similar triangles are triangles whose corresponding angles are congruent and whose corresponding sides are proportional in length. (Definition) Know that a ratio is the relation between two quantities expressed as the quotient of one divided by the other. The ratio of a to b is. (Definition) Know that the ratio of the corresponding sides of similar triangles is called the scale factor. (Definition) Know that a proportion is an equation that states that two ratios are equal. is a proportion. The values a and d are called the extremes of the proportion; b and c are called the means. Know that the product of the extremes equals the product of the means. That is, if ad = bc., then This is called the cross product property.
14 Example Given that and are similar, fill in the blanks. Answer and WorkedOut Solution Since and are corresponding sides and AB is three times DE, and since the corresponding sides of similar triangles are proportional, you know that every side of is three times the corresponding side of. Since,. Since,. (Postulate) Know that if two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar. This is called the AA similarity postulate. (Theorem) Know that if the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. This is called the SSS similarity theorem. (Theorem) Know that if an angle of one triangle is congruent to an angle of another triangle and if the lengths of the sides including these angles are proportional, then the triangles are similar. This is called the SAS similarity theorem. (Theorem) Know that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. This is called the isosceles triangle theorem. (Theorem) Know that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. A corollary is that equilateral triangles are also equiangular, and vice versa. Another corollary is that each angle of an equilateral triangle measures 60. (Theorem) Know that the longest side of a triangle is always opposite the angle with the largest measure. The shortest side of a triangle is always opposite the angle with the smallest measure.
15 (Definition) Know that a median of a triangle is a line or segment that passes through a vertex and bisects the side opposite that vertex. The medians are shown for the triangle above. (Theorem) Know that the medians of a triangle share a common point of intersection. The point is called the centroid. (Definition) Know that an altitude of a triangle is a line or segment through a vertex and perpendicular to the line containing the opposite side. The altitudes are shown for the triangle above. (Theorem) The altitudes of a triangle share a common point of intersection. The point is called the triangle s orthocenter. The angle bisectors are shown for the triangle above. The angle bisectors of a triangle are the bisectors of the triangle s angles. See the definition for the bisector of an angle above.
16 (Theorem) The angle bisectors of a triangle share a common point of intersection. The point is called the triangle s incenter. The perpendicular bisectors are shown for the triangle above. The perpendicular bisectors of a triangle are the perpendicular bisectors of its sides. See the definition for the perpendicular bisector of a line above. (Theorem) Know that the perpendicular bisectors of a triangle share a common point of intersection. The point is called the circumcenter.
17 Key Terms and Concepts for Unit 4 Unit 4 Key Terms triangle triangle angle bisector theorem area cosine HA theorem HL theorem LA theorem LL theorem perpendicular bisector theorem Pythagorean theorem Pythagorean triple sine tangent trigonometric ratios (Definition) Know that the area of a polygon is the number of square units contained in its interior. The formula for the area of a rectangle with height a and base length b is area = b h. The formula for the area of a triangle is area = altitude perpendicular to the base.). (The height is always the length of the (Theorem) Know that in a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. This is called the Pythagorean theorem. (Theorem: converse of the Pythagorean theorem) Know that if the sum of the squares of the measures of two shorter sides of a triangle equals the square of the longest side, then the triangle is a right triangle. (Definition) Know that a Pythagorean triple is a set of three whole numbers a, b, and c that satisfy the equation a 2 + b 2 = c 2. (Theorem) Know that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. This is called the HL theorem. (Theorem) Know that if the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. This is called the LL theorem.
18 Example What is the measure of below? Answer WorkedOut Solution By the LL theorem, the two triangles are congruent. 67 in the other triangle, so by CPCT you know that. corresponds to the angle with measure (Theorem) Know that if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the triangles are congruent. This is called the HA theorem. (Theorem) Know that if one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. This is called the LA theorem. (Theorem) Know that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. This is called the perpendicular bisector theorem. (Theorem) Know that if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. This is called the angle bisector theorem. (Theorem) Know that if a point is in the interior of an angle and equidistant from the two sides of the angle, then it lies on the bisector of the angle. This is the converse of the angle bisector theorem. (Theorem) Know that if the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other. (Theorem) Know that if the length of one leg of a triangle is s, then the length of the other leg is s and the length of the hypotenuse is.
19 Example What is the length of the hypotenuse of this triangle? Answer 2 (Theorem) Know that if the length of the short leg of a triangle is s, then the length of the other leg is and the length of the hypotenuse is 2s. The following are the Trigonometric ratios: (Definition) Know that the sine of an angle is the ratio of the opposite leg length to the hypotenuse length. (Definition) Know that the cosine of an angle is the ratio of the adjacent leg length to the hypotenuse length. (Definition) Know that the tangent of an angle is the ratio of the opposite leg length to the adjacent leg length. Copyright 2012 Apex Learning Inc. (See Terms of Use at
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