Jan Lui Adv Geometry Ch 3: Congruent Triangles 3.1 What Are Congruent Figures? Congruent triangles/polygons : All pairs of corresponding parts are congruent; if two figures have the same size and shape. ABC DEF Reflexive Property: Any segment/angle is congruent to itself. 3.2 Three Ways To Prove Triangles Congruent SSS: If all corresponding sides are congruent, triangles are congruent. SAS: If two sides and an included angle are congruent, triangles are congruent. ASA: If two angles and an included side are congruent, triangles are congruent. SSS SAS ASA 1
3.3 CPCTC and Circles CPCTC : Corresponding Parts of Congruent Triangles are Congruent ABC DEF; <A <D Circles: Circle O, Radius OA All radii of the same circle are congruent. 3.4 Beyond CPCTC Median : line segment drawn from vertex to midpoint of opposite segment Altitude : line segment drawn perpendicular to opposite side Auxiliary lines: additional line segments drawn if necessary Steps Beyond CPCTC : If you prove two triangles congruent through use of one of the methods in 3.3, you can use congruent parts to prove medians, angle bisectors, etc. 2
3.5 Overlapping Triangles Sometimes there is not enough information given to prove two distinct triangles congruent; therefore you must prove that overlapping AEC and ABD are congruent. 3.6 Types of Triangles Scalene: No two sides are congruent Isosceles: At least two sides are congruent Congruent sides are called the legs, the congruent angles between the legs and the base are called base angles, the angle between the legs is called the vertex angle Equilateral: All sides are congruent Equiangular: All angles are congruent Acute: all angles are acute Right: one right angle Segment opposite the right angle is called the hypotenuse, the sides that form the right angle are called the legs Obtuse: one angle is obtuse Scalene Isosceles Equilateral, Equiangular Right Acute Obtuse 3
3.7 Angle Side Theorems If two sides of a triangle are congruent, the base angles are congruent If two angles are congruent, the sides opposite the angles are congruent Larger angle longer side Smaller angle shorter side 3.8 The HL Postulate If the hypotenuse and one leg of a right triangle are congruent to corresponding parts of another right triangle, the two triangles are congruent. ABC DEF 4
Ch 5: Parallel Lines and Related Figures 5.1 Indirect Proof Assume the opposite of what you are trying to prove Prove it until you get a result that directly contradicts what you are given Because your assumption is false, therefore the opposite must be true Assume: <S <T Statement Reason 1) RST is isos w/ vertex R 1) Given 2) RS RT 2) Legs of isos 3) <S <T 3) Because step 3 directly contradicts the given, our assumption is false; therefore <S <T must be true. 5.2 Proving That Lines Are Parallel Exterior Angle Theorem: The measure of an exterior angle is greater than the measure of either remote interior angles. 5
Identifying Parallel Lines: If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel. (Alt int < parallel lines) If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel. (Alt ext < parallel lines) If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel. (Corr < parallel lines) If two lines are cut by a transversal such that two interior angles on the same side are supplementary, the lines are parallel. If two lines are cut by a transversal such that two exterior angles on the same side are supplementary, the lines are parallel. IF: THEN: the lines are If two coplanar lines are perp to a third line, they are parallel 5.3 Congruent Angles Associated With Parallel Lines Parallel Postulate: Through a point not on a line, there is exactly one parallel to the given line. Angles Formed When Parallel Lines Are Cut By Transversal If two parallel lines are cut by a transversal, then any pair of angles formed are either congruent or supp. If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent. ( lines Alt int < ) 6
If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent. ( lines Alt ext < ) If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. ( lines Corr < ) If two parallel lines are cut by a transversal, each pair of interior angles on the same side are supplementary. If two parallel lines are cut by a transversal, each pair of exterior angles on the same side are supplementary. In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. If two lines are parallel to a third line, they are parallel to each other. 7
5.4 Four Sided Polygons Polygons: Not Polygons: Convex Polygon: Each interior angle is less than 180 Diagonals: A line segment that connects two nonconsecutive angles of a polygon Quadrilaterals: Four sided polygon Parallelogram: Both pairs of opposite sides parallel Rectangle: Parallelogram with at least one right angle Rhombus: Parallelogram with four congruent sides Kite: Quadrilaterals with two disjoint pairs of consecutive sides congruent Square: Rectangle and rhombus Trapezoid: Exactly one pair of of parallel sides, called the bases Isosceles Trapezoid: Trapezoid with legs congruent 8
5.5 Properties of a Quadrilateral Properties of Parallelograms Opposite sides parallel and congruent Opposite angles congruent Diagonals bisect each other Consecutives angles supplementary Properties of Rectangles All properties of parallelogram All right angles Diagonals congruent Properties of Kites Two disjoint pairs of consecutive sides congruent Perpendicular diagonals One diagonal is the perp bis. of the other One of the diagonals bisects a pair of opposite angles One pair of opposite angles are congruent Properties of Rhombuses All properties of parallelogram All properties of kite All sides congruent Diagonals bisect the angles Diagonals are perp bisectors of each other Diagonals divide rhombus into four congruent right triangles Properties of Squares All properties of rectangle All properties of rhombus Diagonals form four isos right triangles (45 45 90) Properties of Isosceles Trapezoids Legs are congruent Bases are parallel Lower and upper base angles congruent to each other Diagonals congruent Lower base angles supplementary to upper base angles 9
5.6 Proving That a Quadrilateral is a Parallelogram If both sides of opposite sides are parallel, then a quadrilateral is a parallelogram If both sides of opposite sides are congruent, then a quadrilateral is a parallelogram If one pair of opposite sides is both parallel and congruent, then a quadrilateral is a parallelogram If diagonals bisect each other, then a quadrilateral is a parallelogram If both pairs of opposite angles are congruent, then a quadrilateral is a parallelogram 5.7 Proving That Figures Are Special Quadrilaterals Proving that a Quadrilateral is a Rectangle If a parallelogram contains at least one right angle, then a quadrilateral is a rectangle If the diagonals of a parallelogram are congruent, then a quadrilateral is a rectangle If all four angles of a quadrilateral are right angles, then a quadrilateral is a rectangle Proving that a Quadrilateral is a Kite If two disjoint pairs of consecutive sides of a quadrilateral are congruent, it is a kite If one of the diagonals is a perpendicular bisector of the other, it is a kite Proving that a Quadrilateral is a Rhombus If a parallelogram contains a pair of consecutive congruent sides, it is a rhombus If either diagonal bisects two angles, it is a rhombus If diagonals are perp bisectors of each other, it is a rhombus Proving that a Quadrilateral is a Square If it is both a rectangle and a rhombus then it is a square Proving that a Trapezoid is Isosceles If legs of a trapezoid are congruent, then it is isosceles If the lower base or upper base angles are congruent, then it is isosceles If diagonals are congruent, it is isosceles 10
Ch 9: The Pythagorean Theorem 9.1 Review of Radicals and the Quadratic Equations ab = a b a 2 x + b x x yx = y Solve x 2 + 9 = 25 2 = a2 x + b 2 x = a x + b x = ( a + b ) x Method 1: Method 2: Factoring x 2 + 9 = 25 x 2 + 9 = 25 x 2 = 16 x 2 16 = 0 x = ± 4 ( x 4 )(x + 4 ) = 0 x = 4, 4 9.2 Introduction to Circles Circumference : π d Area : πr 2 Arc : made up of two points on a circle and all points in between them Measure of an arc: equivalent to the measure of degrees it occupies Length: fraction of the circumference l = measure 360 C Sector : region bounded by two radii and an arc a = measure π 360 r 2 Chord : line segment joining two points on a circle Inscribed angle: vertex on circle, sides determined by two chords 11
9.3 Altitude Hypotenuse Theorems If an altitude is drawn to the hypotenuse of a right triangle: ABC~ ACD~ CBD h 2 = x y b 2 = x c a 2 = y c 9.4 Geometry s Most Elegant Theorem Pythagorean Theorem : a 2 + b 2 = c 2 9.5 The Distance Formula To find the distance between 2 points: (x 1,y 1 ) and (x 2,y 2 ) 12
9.6 Families of Right Triangles Pythagorean Triple : Any three whole numbers that satisfy the Pythagorean Theorem ie: (3,4,5) (5,12,13) (7,24,25) (8,15,17) (9,40,41) etc The Principle of Reduced Triangles: 9.7 Special Right Triangles 30 60 90 Triangle: 45 45 90 Triangle: 13
9.8 The Pythagorean Theorem and Space Figures Face: ABFE Edge: AB Diagonal: HB 9.9 Introduction to Trigonometry Base : JOMK Vertex : P Altitude : PR (perpendicular to center of base) Slant height : PS (perpendicular to side of base) opposite leg hypotenuse adjacent leg hypotenuse opposite leg adjacent leg sin of <A:, cos of <A:, tan of <A:, a c b c 9.10 Trigonometric Ratios b a 14
Practice Problems: 1. Statement Reason 1) LMN is an isosceles triangle with vertex M. MP bisects LN 1) Given 2) ML MN 2) Legs of isos 3) <N <L 3) 4) LP PN 4) Def of seg bisector 5) LMP NMP 5) SAS (2,3,4) 6) <LMP <NMP 6) CPCTC 2. Statement Reason 1) ZX YW <YXW and <ZWX are rt 1) Given 2) WX WX 2) Reflex 3) ZWX YXW 3) HL (1,1,2) 4) <Z <Y 4) CPCTC 15
3. Assume that BC DC. Statement Reason 1) AB AD <BAC <DAC 1) Given BC DC 2) AC AC 2) Reflex 3) ADC ABC 3) SSS (1,1,2) 4) <BAC <DAC 4) CPCTC Our assumption is false because <BAC <DAC directly contradicts the given; therefore, BC DC. 4. <ACB <2, def of isos <ACB <3, vertical <s <2 <3, trans prop <1 is supp to <2, 2 adj < that form straight line are supp <1 is supp to <3, trans prop 7x + 15 + 4x =180 11x + 15 = 180 11x = 165 x=15 <1 = 7(15) + 15 = 120 180 120 = 60 <2 = 60 16
5. P: NP RS Statement Reason 1) Diagram as shown 1) Given 2) ON OR 2) Radii of same circle are congruent 3) <ONR <ORN 4) <ONR <P <ORN <S 3) 4) lines corr <s 5) <P <S 5) Trans Prop 6) OS OP 7) OP ON = NP OS OR = RS 6) 7) Seg Subt Post 8) NP RS 8) seg seg = seg 6. Solve for x. What is the perimeter of the kite? 2x 9 = 29 x 3 3 (2x 9) = 29 x 6x 27 = 29 x 7x = 56 x= 8 3(8) + 6 2(8) 9 +8 +7 + + 29 8 = 16 9 +8 +7 + 24+6 + 21 = 22 + 30 +7 = 22 +15 +7 = 44 units 2 3 2 3 2 17
7. Statement Reason 1) Given 1) 2) AD CB 2) Opp sides of parallelogram are 3) ADF CBE 3) SAS (1, 1, 2) 4) <DFA <CEB 4) CPCTC 5) DF EB 5) Alt ext <s lines 8. Find the area of the blue region to the nearest whole number: r = 2.5 m (blue) = 180 75 = 105 105 π(2.5) 360 2 = 5.72 6 units 18
9. a 2 = 5 9 = 45 a = 45 = 3 5 b 2 = 5 ( 5 + 9 ) = 70 b = 70 c 2 = 9 ( 5 + 9 ) = 126 c = 126 = 3 14 10. Find the length of the altitude to the base of ABC if <A = 120, AB AC, and AB = 10. ABD is a 30 60 90 triangle because AD bisects <BAC. AB = 2x, AD = x, BD = x 3 AB = 2x = 10, then x = 5 AD = 5 units 19