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Transcript
Chapters 4 and 5 Triangles Which type of triangle is this? Scalene Triangle None of the sides and none of the angles are the same. Which type of triangle is this? Isosceles Triangle Two sides and two angles are the same. Which type of triangle is this? Equilateral Triangle All sides and all angles are the same. Opposite A Which side is opposite โ ๐จ? ๐ฉ๐ช Which side is opposite โ ๐ฉ? ๐จ๐ช Which side is opposite โ ๐ช? ๐จ๐ฉ C B Congruent Triangles 4.1 Theorem 4.1: Triangle Sum Theorem- The sum of all the interior angles of every triangle is ๐๐๐°. 60° 15° 100° 60° 60° 65° Practice Solve for x and y. ๐ ๐๐ โ ๐๐ + ๐๐ โ ๐ + ๐๐ = ๐๐๐ โ๐๐ โ๐๐ ๐๐ โ ๐๐ + ๐๐ โ ๐ = ๐๐ ๐๐๐ โ ๐๐ = ๐๐ +๐๐ +๐๐ ๐๐๐ = ๐๐๐ ๐๐ ๐๐ ๐๐ โ ๐๐ ๐=๐ ๐๐ โ ๐ ๐๐ โ ๐๐ + ๐ = ๐๐๐ ๐ ๐ โ ๐๐ + ๐ = ๐๐๐ ๐๐ + ๐ = ๐๐๐ โ๐๐ โ๐๐ ๐ = ๐๐๐ Theorem 4.2: Triangle Exterior Angle Conjecture- The measure of the exterior angle of a triangle isโฆ equal to the sum of the measures of the two nonadjacent interior angles. ๐๐° ๐ The angles circled in red are nonadjacent interior angles for x. ๐๐° ๐๐° ๐ = ๐๐๐° Which side is opposite โ ๐จ๐ฉ๐ซ ? ๐จ๐ซ A D Which side is opposite โ ๐ฉ๐ซ๐ช ? ๐ฉ๐ช Which angle is opposite ๐จ๐ฉ ? โ ๐จ๐ซ๐ฉ B C Practice Find the missing angle measures. ๐๐๐ โ ๐๐ = ๐๐ ๐๐๐ = ๐๐ ๐ ๐ ๐๐. ๐ = ๐ ๐๐° Practice Find the missing angle measure. x ๐ = ๐๐ Which type of triangle is this? ๐๐° Acute Isosceles Which type of triangle is this? 9 CM 11 CM 7 CM Right Scalene Practice Find the degree measure of the interior angles of โ๐พ๐๐. โ ๐พ = (๐ + ๐๐) โ ๐พ = (๐๐ + ๐๐) โ ๐พ = ๐๐° โ ๐ = (๐๐ + ๐๐) โ ๐ = ๐(๐๐) + ๐๐ โ ๐ = ๐๐° โ ๐ = (๐๐ + ๐) โ ๐ = ๐ ๐๐ + ๐ โ ๐ = ๐๐° ๐ + ๐๐ + ๐๐ + ๐๐ + ๐๐ + ๐ = ๐๐๐ ๐๐ + ๐๐ = ๐๐๐ โ๐๐ โ๐๐ ๐๐ = ๐๐๐ ๐ ๐ ๐ = ๐๐ How do we know our answer is correct? Three angles in any triangle should add up to ๐๐๐° โ ๐พ = ๐๐° + โ ๐ = ๐๐° + โ ๐ = ๐๐° ๐๐๐° Practice Find the degree measure of the interior angles of โ๐จ๐ฉ๐ช. โ ๐จ = (๐๐ + ๐) โ ๐จ = ๐(๐) + ๐) โ ๐จ = ๐๐° โ ๐ฉ = (๐๐๐ โ ๐๐) โ ๐ช = (๐๐๐ โ ๐๐) โ ๐ฉ = ๐๐ ๐ โ ๐๐ โ ๐ช = ๐๐ ๐ โ ๐๐ โ ๐ฉ = ๐๐° โ ๐ช = ๐๐° ๐๐ + ๐ + ๐๐๐ โ ๐๐ + ๐๐๐ โ ๐๐ = ๐๐๐ ๐๐๐ โ ๐๐ = ๐๐๐ +๐๐ +๐๐ ๐๐๐ = ๐๐๐ ๐๐ ๐๐ ๐=๐ How do we know our answer is correct? Three angles in any triangle should add up to ๐๐๐° โ ๐จ = ๐๐° + โ ๐ฉ = ๐๐° + โ ๐ช = ๐๐° ๐๐๐° Practice Find the degree measure of the interior angles of triangle ABC. โ ๐จ = (๐ + ๐) โ ๐ฉ = (๐๐ + ๐) โ ๐ช = (๐๐ + ๐) Review Are the lines given by the following equations perpendicular, parallel, or neither? ๐ = ๐๐ โ ๐๐ ๐ ๐=โ ๐+๐ ๐ Perpendicular Review Are the lines given by the following equations perpendicular, parallel, or neither? ๐ = ๐๐ โ ๐๐ ๐ = ๐๐ + ๐ Parallel Review Are the lines given by the following equations perpendicular, parallel, or neither? ๐ ๐= ๐โ๐ ๐ Neither ๐ ๐= ๐+๐ ๐ Review Are the lines given by the following equations perpendicular, parallel, or neither? ๐๐ + ๐๐ = ๐ ๐ ๐= ๐+๐ ๐ ๐๐ โ ๐๐ = ๐ ๐ ๐=โ ๐+๐ ๐ Perpendicular Congruent Triangles 4.2 Congruent Triangles: When two triangles are the exact same size and shape they are said to be congruent. Even though they have the same shape and size, they may be positioned differently. 4.2 What corresponding parts of the two congruent triangles are congruent? z a y b x c ๐๐ โ ๐๐ โ ๐ โ โ ๐ ๐๐ โ ๐๐ โ ๐ โ โ ๐ ๐๐ โ ๐๐ โ ๐ โ โ ๐ โ๐๐๐ โ โ๐๐๐ This slide demonstrates the concept of CPCTC. If two triangles are congruent, then the corresponding parts of those congruent triangles are congruent. z a y b x c ๐๐ โ ๐๐ โ ๐ โ โ ๐ ๐๐ โ ๐๐ โ ๐ โ โ ๐ ๐๐ โ ๐๐ โ ๐ โ โ ๐ Congruent Triangles 4.2 Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angle must also be congruent. ๐๐° ๐๐° ๐๐° ๐๐° ๐๐° ๐๐° 4.2 Isosceles Triangles Isosceles Triangle Conjecture: An isosceles triangle has two congruent angles. The sides opposite the congruent angles(legs) are also congruent. Vertex Angle 4.2 Isosceles Triangles Converse of the Isosceles Triangle Conjecture: A triangle that has two congruent angles must be isosceles. If a triangle has two congruent sides, then it is isosceles. Vertex Angle 4.2 Isosceles Triangles Vertex Angle: Angle between the two congruent sides. Legs: Congruent sides. Base Angle Base Angle Base 4.3 By comparing only three parts of two different triangles we will try to determine if the two triangles are congruent. Basically, we will try to answer the following question: Is โ ? 4.4 You have already seen that if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Which conjecture tells you this? The Third Angle Conjecture 60 ฬ 60 ฬ 90 ฬ 90 ฬ 30 ฬ 30 ฬ Just what is this congruence you speak of? z a y b x c ๐๐ โ ๐๐ S โ ๐๐๐ โ โ ๐๐๐ A ๐๐ โ ๐๐ S โ๐๐๐ โ โ๐๐๐ 4.3 Side-Side-Side (SSS) Congruence Postulate: If three sides in one triangle are congruent to corresponding sides in another triangle, then the triangles are congruent. d s e t w How do we indicate their congruence? โ๐๐๐ โ โ๐ ๐๐ h 4.3 Side-Angle-Side (SAS) Congruence Postulate: If corresponding sides and the included angle (angle between the two congruent sides) are congruent in two triangles, then the triangles are congruent. w z t y x How do we indicate their congruence? โ๐๐๐ โ โ๐๐๐ s 4.4 Angle-Side-Angle (ASA) Congruence Postulate: If corresponding angles and the included side (the side between the angles) are congruent in two triangles, then the triangles are congruent. x b y c a How do we indicate their congruence? โ๐๐๐ โ โ๐๐๐ z 4.4 Side-Angle-Angle (SAA) Congruence Theorem: If corresponding angles and a non-included side of two triangles are congruent, then the triangles are congruent. x b y c a How do we indicate their congruence? โ๐๐๐ โ โ๐๐๐ z 4.6 Hypotenuse-Leg (HL) Congruence Theorem: If the hypotenuse and a corresponding leg of two right triangles are congruent, then the triangles are congruent. A B D C E How do we indicate their congruence? โ๐จ๐ฉ๐ช โ โ๐ซ๐ฌ๐ญ F 4.4 Side-Side-Angle (SSA) Congruence Conjecture: If corresponding sides and a non-included angle of two triangles are congruent, then the triangles are congruent. x b y c a not enough by itself z 4.4 Angle-Angle-Angle (AAA) Congruence Conjecture: If two triangles have all congruent angles, then the triangles are congruent. not enough by itself Are the two triangles congruent? If so, how do you know? SAS Are the two triangles congruent? If so, how do you know? SSS Are the two triangles congruent? If so, how do you know? Not Enough Information Are the two triangles congruent? If so, how do you know? SAS Are the two triangles congruent? If so, how do you know? SSS Are the two triangles congruent? If so, how do you know? AAS Are the two triangles congruent? If so, how do you know? AAS Are the two triangles congruent? If so, how do you know? ASA Are the two triangles congruent? If so, how do you know? HLT Are the two triangles congruent? If so, how do you know? SAA Are the two triangles congruent? If so, how do you know? SAS Practice For the following slides, complete the two column proofs. Statements Prove โ ๐จ โ โ ๐ช Reasons 1. ๐จ๐ซ โ ๐ซ๐ช, 1. Given โ ๐จ๐ซ๐ฉ โ โ ๐ช๐ซ๐ฉ A D 2. ๐ฉ๐ซ โ ๐ฉ๐ซ 2. Reflexive property 3. โ๐จ๐ซ๐ฉ โ โ๐ช๐ซ๐ฉ 3. SAS 4. โ ๐จ โ โ ๐ช 4. CPCTC B C Statements Prove: โ ๐ท๐ธ๐บ โ โ ๐น๐ธ๐บ Reasons 1. ๐ท๐ธ โ ๐ธ๐น, 1. Given ๐ท๐บ โ ๐น๐บ Q 2. ๐ธ๐บ โ ๐ธ๐บ 2. Reflexive property 3. โ๐ท๐ธ๐บ โ โ๐น๐ธ๐บ 3. SSS 4. โ ๐ท๐ธ๐บ โ โ ๐น๐ธ๐บ 4. CPCTC P R S Statements Prove ๐จ๐ฉ โ ๐ซ๐ฌ A D Reasons 1. ๐จ๐ซ โฅ ๐ฉ๐ฌ, 1. Given โ ๐จ โ โ ๐ฌ 2. ๐ฉ๐ซ โ ๐ฉ๐ซ 2. Reflexive property 3. โ ๐จ๐ซ๐ฉ โ โ ๐ฌ๐ฉ๐ซ 3. Alternate Interior โ s Theorem 4. โ๐จ๐ซ๐ฉ โ โ๐ฌ๐ฉ๐ซ 4. AAS B E 5. ๐จ๐ฉ โ ๐ซ๐ฌ 5. CPCTC Theorem 4.6: Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. D F E If ๐ซ๐ญ โ ๐ฌ๐ญ, ๐๐๐๐ โ ๐ซ โ โ ๐ฌ. Theorem 4.7: Converse of the Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent. D F E If โ ๐ซ โ โ ๐ฌ, ๐๐๐๐ ๐ซ๐ญ โ ๐ฌ๐ญ. Practice What is the length of ๐ฉ๐ช? A ๐๐๐๐ C B ๐ฉ๐ช = ๐๐๐๐ฆ Practice What is the degree measure of โ ๐ซ? D ๐๐° ๐๐๐๐ ๐๐๐ โ ๐๐ = ๐๐๐ ๐๐° C ๐๐๐ ÷ ๐ = ๐๐ โ ๐ซ = ๐๐° ๐๐° E ๐๐๐๐ Corollaries Corollary to Theorem 4.6 If a triangle is equilateral, then it must be equiangular. Corollaries Corollary to Theorem 4.7 If a triangle is equiangular, then it must be equilateral. Practice A What is the degree measure of โ ๐ฉ? ๐๐° ๐๐๐๐ ๐๐๐๐ ๐๐๐ ÷ ๐ = ๐๐ โ ๐ฉ = ๐๐° B ๐๐° ๐๐° ๐๐๐๐ C 5.1 Bisectors of a Triangle Any point on the perpendicular bisector of a segment must be equidistant from the endpoints of the segment. A B C ๐จ๐ฉ โ ๐จ๐ซ D 5.1 Bisectors of a Triangle Concurrency of Perpendicular Bisectors of a Triangle: Perpendicular bisectors of a triangle are all concurrent at a point called the circumcenter. This point is equidistant from the all the vertices in the triangle. A E F G B D ๐จ๐ฎ = ๐ฉ๐ฎ = ๐ช๐ฎ C 5.3 Median A median goes from a vertex in a triangle to the midpoint of the opposite side. A E F G B D C 5.3 Median What are the medians in this triangle? ๐ญ๐ช, ๐ฉ๐ฌ, and ๐จ๐ซ are medians. A E F G B D C 5.3 Median The three medians of a triangle are all concurrent at a point called the centroid. A E F G B Point ๐ฎ is the centroid. D C Theorem 5.7 Concurrency of Medians of a Triangle The centroid is located at two thirds of the distance from each vertex to the midpoint to the opposite side. If ๐ฎ is the centroid โ๐จ๐ฉ๐ช, ๐ญ๐ก๐๐ง: ๐ ๐จ๐ฎ = ๐จ๐ซ ๐ A E F ๐ ๐ช๐ฎ = ๐ช๐ญ ๐ ๐ ๐ฉ๐ฎ = ๐ฉ๐ฌ ๐ G B D C ๐ฎ is the centroid of โ๐จ๐ฉ๐ช. A 1. If ๐จ๐ฎ = ๐๐๐๐, what is the Length of ๐จ๐ซ ? ๐จ๐ซ = ๐๐๐๐ 2. If ๐ญ๐ฎ = ๐๐๐, what is the Length of ๐ญ๐ช ? ๐ญ๐ช = ๐๐๐๐ 3. If ๐ฉ๐ฌ = ๐๐๐๐, what is the length of ๐ฉ๐ฎ ? ๐ฉ๐ฎ = ๐๐๐๐ E F G B D C 5.3 Altitude An altitude in a triangle goes from a vertex and creates a right angle (is perpendicular with) the opposite side. An altitude can lie in, on, or outside the triangle. X B W Y Z R S A T D C 5.3 Altitude All of the altitudes in a triangle are concurrent at a single point called the orthocenter. 5.4 Midsegment: segment connecting midpoints of two sides of a triangle. A ๐ฎ๐ฉ, ๐ฉ๐ซ, and ๐ซ๐ฎ are all midsegments. G E B D C 5.4 ๐ ๐ Midsegment Theorem: a midsegment is the length of the side of the triangle it is parallel to. A G ๐ ๐ฎ๐ฉ = ๐ฌ๐ช ๐ ๐ ๐ฉ๐ซ = ๐จ๐ฌ ๐ B ๐ ๐ฎ๐ซ = ๐จ๐ช ๐ E D C Practice ๐ญ๐ฉ, ๐ฉ๐ซ, and ๐ซ๐ญ are all midsegments. Find the lengths of segments ๐จ๐ช, ๐ฉ๐ซ, and ๐ช๐ฌ. B A C 4 ๐จ๐ช = ๐๐ 11 F 5 D ๐ฉ๐ซ = ๐. ๐ ๐ช๐ฌ = ๐ E Practice ๐ฎ๐ฑ = Midsegment G R T J ๐ฉ๐ซ = Altitude B S ๐ฟ๐พ = Median X A D Z W C Y 5.5 Theorem 5.10 Side-Angle Inequality Theorem: The longest side in a triangle is opposite the largest angle. The shortest side in a triangle is opposite the smallest angle. The side with length in between the lengths of the other sides is opposite the angle that has degree measure in between the other angles. ๐๐๐ ๐๐° ๐๐° ๐๐๐ ๐๐๐ ๐๐° Side-Angle Inequality B ๐๐๐° ๐๐° ๐๐° A C List the sides in order from longest to shortest. ๐จ๐ช > ๐จ๐ฉ > ๐ฉ๐ช 5.5 Theorem 5.13 Triangle Inequality: The sum of the length of two sides of a triangle must be greater than the length of the third side. B 8cm 10cm A C 17cm ๐๐ + ๐ > ๐๐ ๐๐ + ๐๐ > ๐ โ๐จ๐ฉ๐ช could be a triangle. ๐ + ๐๐ > ๐๐ 5.5 Theorem 5.13 Triangle Inequality: The sum of the length of two sides of a triangle must be greater than the length of the third side. B 8cm 10cm A C 20cm โ๐จ๐ฉ๐ช could not be a triangle. ๐๐ + ๐ > ๐๐ Practice Determine whether it is possible to draw a triangle with the sides of the given measurements. ๏ 8cm, 11cm , 18.5cm No ๏ 7cm, 17cm , 10.5cm Yes ๏ 9cm, 6cm , 19cm No How do you know? 5.6 Hinge Theorem ๏ Sometimes a picture is worth a thousand confusing math terms. ๏ Which side do you predict will be longer ๐จ๐ฉ or ๐ซ๐ฌ ? C ๐จ๐ฉ < ๐ซ๐ฌ ๐๐° A B F ๐๐๐° E D Practice Fill in the blank with a โless thanโ, โgreater thanโ, or โequal toโ symbol. ๐ช๐ฉ > ๐ญ๐ฌ C F B ๐๐° E A ๐๐° D Practice For the following slides, complete the two column or flow chart proofs. Practice Prove that the two triangles are congruent Statements a Reasons b 1. โ ๐๐๐ โ โ ๐ ๐๐ 1. Given 2. ๐๐ โ ๐๐ 2. Reflexive property 3. โ ๐๐๐ โ โ ๐ ๐๐ 3. Alternate Interior โ s Theorem 4. โ๐๐๐ โ โ๐ ๐๐ 4. ASA c d Prove that ๐๐ โ ๐๐. Statements w c Reasons 1. Given 1. ๐๐ โ ๐๐, ๐๐ โ ๐๐ 2. โ ๐๐๐ โ โ ๐๐๐ 2. Vertical Angles Theorem n 3. โ๐๐๐ โ โ๐๐๐ 3. SAS r 4. ๐๐ โ ๐๐ o 4. CPCTC Given: ๐ฉ๐ช โ ๐ช๐ซ, ๐จ๐ฉ ll ๐ซ๐ฌ Prove: ๐จ๐ช โ ๐ช๐ฌ A D C B E Flow Chart a d Prove โ ๐ โ โ ๐ b c e f ๐๐ โ ๐ ๐ Given โ ๐ โ โ ๐ โ๐๐๐ โ โ๐ ๐๐ โ ๐ โ โ ๐ Given AAS CPCTC โ ๐ โ โ ๐ Right โ Congruence Theorem a ๐๐ is an altitude of โ๐๐๐ Prove โ ๐ โ โ ๐ c b d ๐๐ โ ๐๐ Given โ ๐๐๐ โ โ ๐๐๐ โ๐๐๐ โ โ๐๐๐ โ ๐ โ โ ๐ Definition of an altitude SAS CPCTC ๐๐ โ ๐๐ Reflexive Property Fill in the blanks. Prove โ ๐ฉ โ โ ๐ซ. Given: C is the midpoint of ๐ฉ๐ซ, ๐จ๐ฉ โ ๐จ๐ซ A Statements 1. ๐จ๐ฉ โ ๐จ๐ซ 1. Given 2. ๐จ๐ช โ ๐จ๐ช 2. Reflexive Property 3. Definition of a midpoint 3. ๐ฉ๐ช โ ๐ช๐ซ 4. โ๐จ๐ฉ๐ช โ โ๐จ๐ซ๐ช B C Reasons D 5. โ ๐ฉ โ โ ๐ซ 4. SSS 5. CPCTC Prove angle B is congruent to angle D. ๐จ๐ช is a median of triangle ABD. A Statements B C D Reasons Given: ๐ฟ๐ โ ๐ฟ๐พ, ๐๐พ โฅ ๐ฟ๐ Prove โ ๐ โ โ ๐พ X Statements 1. ๐ฟ๐ โ ๐ฟ๐พ, ๐๐พ โฅ ๐ฟ๐ Reasons 1. Given 2. โ ๐ฟ๐๐ โ โ ๐ฟ๐๐พ 2. Definition of โฅ lines 3. ๐ฟ๐ โ ๐ฟ๐ 3. Reflexive Property 4. โ๐ฟ๐๐ โ โ๐ฟ๐๐พ 4. HLT 5. โ ๐ โ โ ๐พ 5. CPCTC Y Z W Practice ๐+๐ ๐โ๐ 23 Practice x 62 ฬ y x 110 ฬ 44 ฬ 135 ฬ 77 ฬ z