45 Wyner Math Academy I Spring 2016

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1 45 Wyner Math cademy I Spring 2016 HPTER FIVE: TRINGLES Review January 13 Test January 21 Other than circles, triangles are the most fundamental shape. Many aspects of advanced abstract mathematics and practical applications are based on properties of triangles. 5- Types and omponents of Triangles Monday 1/4 isosceles equilateral scalene median altitude point of concurrency incenter circumcenter orthocenter centroid ➊ Identify types of triangles. ➋ onstruct the medians of a triangle. ➌ onstruct the altitudes of a triangle. ➍ Find the incenter, circumcenter, orthocenter, and centroid of a triangle. 5- ongruence and Similarity Wednesday 1/6 PT proportion similar ➊ Show that two triangles are congruent. ➋ Show how a pair of SS triangles may or may not be congruent. ➌ Identify the values needed to make a polygon proportional to a given polygon. ➍ Show that two triangles are similar. ➎ Use a two-column format to prove similarity or congruence of two triangles. 5- Trigonometric Functions in Right Triangles Friday 1/8 trigonometric function sine cosine tangent cosecant secant cotangent ➊ Find the values of each of the six trigonometric functions of a nonright angle in a right triangle with two known sides. ➋ Find values of cotangent, secant, and cosecant on the calculator. ➌ alculate a side length in a right triangle based on a known angle and known side length. 5- Inverse Trigonometric Functions in Right Triangles Wednesday 1/13 inverse trigonometric function sin -1 (arcsin) cos -1 (arccos) tan -1 (arctan) ➊ alculate an angle measure in a right triangle based on two known side lengths. ➋ Solve a right triangle.

2 46 Wyner Math cademy I Spring Types and omponents of Triangles If two of the angles are congruent, the two sides opposite them are congruent and the triangle is ISOSELES. If all three angles are congruent, the three sides are congruent and the triangle is EQUILTERL. Otherwise the triangle is SLENE. The largest angle in a triangle determines whether the triangle itself is acute, right, or obtuse. ➊ Identify types of triangles. 1. Identify if any side is congruent to another. If not, the triangle is scalene. If so, the triangle is isosceles, and if all three sides are equal, it is also equilateral. 2. Identify whether the largest angle is acute (< 90 ), right (90 ), or obtuse (> 90 ). ➊ Identify the type of triangle shown at right. 1. Two sides are congruent, so it is isosceles. The third is not, so it is not equilateral. 2. The largest angle is more than 90, so it is obtuse. MEIN of a triangle is a line segment that connects a vertex with the midpoint of the opposite side. n LTITUE of an acute triangle is a line segment that is perpendicular to a side and that passes through a vertex. In an obtuse triangle, two of the altitudes are outside the triangle, and thus do not cross a side but are still perpendicular to the line that contains the side. ➋ onstruct the medians of a triangle. 1. isect each side (see 2-). 2. onnect each midpoint with the vertex opposite it. ➌ onstruct the altitudes of a triangle. 1. If the triangle is obtuse, extend the shorter sides. 2. Through each vertex, construct a line perpendicular to the side opposite it (see 2-). point where three or more lines meet is a Point of ONURRENY. The angle bisectors are concurrent at the INENTER. It is called this because it is the center of a circle inscribed within the triangle. The perpendicular bisectors are concurrent at the IRUMENTER. It is called this because it is the center of a circle circumscribed about the triangle. The altitudes are concurrent at the ORTHOENTER. It is called this because the altitudes are orthogonal with the sides. The medians are concurrent at the ENTROI. It is called this because this would be the center of mass if the triangle were a physical object. ➍ Find the incenter, circumcenter, orthocenter, and centroid of a triangle. 1. To find the incenter, construct the three angle bisectors (see 2-). To find the circumcenter, construct the three perpendicular bisectors (see 2-). To find the orthocenter, construct the three altitudes (see ➌). To find the centroid, construct the three medians (see ➋). 2. Find the point of concurrency of the three lines.

3 47 Wyner Math cademy I Spring ongruence and Similarity Two geometric objects are congruent if every part of the first one is congruent to the corresponding part in the second one. For example, Δ = ΔEF if =, = E, = F, = E, = F, and = EF. The converse is also true: If two geometric objects are congruent, every part of the first is congruent to the corresponding part of the second. For triangles, this is abbreviated PT: orresponding parts of congruent triangles are congruent. This applies to every aspect of the triangles: sides, angles, altitudes, medians, etc. In most cases, if a total of three sides and angles are known to be congruent to their corresponding parts of another triangle, this is sufficient to establish that the two triangles are congruent. orresponding congruent parts bbreviation ongruence all three sides SSS definitely two angles and a side S or S definitely two sides and the angle between them SS definitely two sides and an angle not between them SS definitely if the angle is not acute, otherwise possibly all three angles probably not ➊ Show that two triangles are congruent. 1. Identify a side in the first triangle that is congruent to a side in the second triangle. 2. Identify a second part of the first triangle (side or angle) that is congruent to the corresponding part in the second triangle. 3. Repeat step 2 for a third part. If only one of the three parts is an angle, it must be the angle between the sides. 4. Identify which pattern of congruence you have demonstrated (SS, S, S, or SS). ➊ Given R is the perpendicular bisector of VL, show that ΔVR ΔLR. 1. R R, because it is the same line segment. 2. VR LR, because a bisector cuts a line segment into two equal parts. V 3. VR LR, because the perpendicular makes a 90 angle on each side. 4. SS ➋ Show how a pair of SS triangles may or may not be congruent. 1. Sketch a triangle. Measure the three sides and the smallest angle. 2. In the start of a new triangle, duplicate the measured angle and the sides making it. 3. onnect the ends of the two sides to complete the copy of the triangle. 4. Using the same length as in step 3, sketch a side from the end of the short side to a different point on the long side. ➋ Sketch two different triangles in which = 30, = 30, and = 40. R L

4 48 Wyner Math cademy I Spring 2016 PROPORTION is a ratio. Two variables are proportional if one equals the other multiplied by a constant ratio. ➌ Identify the values needed to make a polygon proportional to a given polygon. 1. Identify the multiplier (ratio) needed to get from a given length in the polygon to the corresponding given length in the other polygon. This can be a fraction where the new length is the numerator and the old length is the denominator. 2. Multiply each other length in the original polygon by this ratio. ➌ Identify the values of a and b in the second triangle at right, given the two triangles are proportional. a 5 7 b 1. The ratio is = a = 5( 8 5) = 8 b = 7( 8 5) = 11.2 Two geometric objects are SIMILR if every angle of the first one is congruent to the corresponding angle of the second one, and every side of the first one is proportional to the corresponding side of the second one. In similar triangles, every length in the first object, such as a triangle s sides, altitudes and medians, is proportional to the corresponding length in the second object. If a pair of triangles has two pairs of congruent angles (), the two triangles are similar. If a pair of triangles has two pairs of proportional sides, the two triangles are similar if the side between them is also proportional (SSS) or if the angle between them is congruent (SS). ➍ Show that two triangles are similar. 1. o one of the following. a) Find the measures of two pairs of corresponding angles. If each pair is congruent, the two triangles are similar. b) Find the measures of two pairs or corresponding sides. If the ratio is the same for the two pairs, the triangles are similar if the angle between them is congruent for the two triangles or if the third pair of sides follows the same ratio as the other pairs. ➍ Show that the and E are similar. 1. a) Two pairs of corresponding angles are congruent: E and E >> >> E

5 49 Wyner Math cademy I Spring 2016 geometric PROOF is a formal, irrefutable, detailed explanation of a claim. Proofs are commonly written in two-column form, with one column for statements based on given information or previous statements and the other column for justification for each statement in the first column. Paragraph form and flowchart form are other ways of showing proofs. ➎ Use a two-column format to prove similarity or congruence of two triangles. 1. In the left column, state the relevant given information. 2. In the right column, state that this information is given. 3. In the left column, state what can be concluded based on the given information. 4. In the right column, give a formal justification (such as an established property or theorem) for each conclusion. 5. To prove congruence, make further conclusions in the left column based on the information above until you have concluded that each side of one triangle is congruent with a corresponding side of the other triangle (SSS), two angles and one side are congruent (S or S), or two sides and the angle between them are congruent (SS). To prove similarity, make further conclusions in the left column based on the information above until you have concluded that two of the angles of one triangle are congruent to two of the angles in the other triangle (), or that two sides of one triangle are proportional two sides of the other triangle and the side between them is also proportional (SSS) or the angle between them is congruent (SS). 6. In the right column, repeat step 4 for the new information. ➎ Prove that Δ is congruent to Δ. statement justification given Δ is isosceles definition of isosceles triangle ase angles of an isosceles triangle are congruent. m = 90 given and are supplementary Linear angles are supplementary. m = 90 and are supplementary transitive property Δ ~ Δ theorem of similarity orresponding angles of similar triangles are congruent. reflexive property Δ Δ SS theorem of congruence oncepts from hapter 2, especially those involving congruent angles, can be useful in proving congruency or similarity, as can other theorems such as those below. TRINGLE SUM Theorem: The interior angles of a triangle have a sum of 0. TRINGLE MISEGMENT Theorem: line segment connecting the midpoints of sides of a triangle is parallel to the third side of the triangle. TRINGLE PROPORTIONLITY Theorem: line through a triangle parallel to one of the sides creates a triangle that is similar to the triangle it is within. The converse is true as well. NGLE ISETOR Theorem: The line segments created by an angle bisector in a triangle are proportional to the corresponding sides of the triangle.

6 50 Wyner Math cademy I Spring Trigonometric Functions in Right Triangles The three primary TRIGONOMETRI Functions are SINE (sin), OSINE (cos), and TNGENT (tan). They show the ratio of the length of one side of a right triangle to the length of another side. opposite sin ø = hypotenuse cos ø = adjacent opposite hypotenuse tan ø = adjacent The three reciprocal trigonometric functions are OSENT (csc), SENT (sec), and OTNGENT (cot). They are the reciprocals of sine, cosine, and tangent, respectively. 1 hypotenuse 1 hypotenuse 1 adjacent sin ø = csc ø = opposite cos ø = sec ø = adjacent tan ø = cot ø = opposite ➊ Find the values of each of the six trigonometric functions of a nonright angle in a right triangle with two known sides. 1. Use the Pythagorean theorem a 2 + b 2 = c 2 to find the length of the third side. 2. Identify which length is opposite the given angle, which is adjacent to it, and which is the hypotenuse. 3. For each trig function, use the two appropriate lengths in a fraction based on the equations above. ➊ Find the sine, cosine, tangent, cosecant, secant, and cotangent of angle shown at right. 1. a = a = 9 = 3 a 2. opposite = 3, adjacent = 4, hypotenuse = 5 3. sin = 3 / 5 cos = 4 / 5 tan = 3 / 4 csc = 5 / 3 sec = 5 / 4 cot = 4 4 / 3 Most calculators do not have buttons for the reciprocal trigonometric functions. ➋ Find values of cotangent, secant, and cosecant on the calculator. 1. Type 1/ and then the reciprocal function. ➋ Evaluate sec sec 25 = 1/cos(25) 1.10 trigonometric function can be used to calculate an unknown side length from a known angle and known side length in a right triangle. ➌ alculate a side length in a right triangle based on a known angle and known side length. 1. Write a sine, cosine, or tangent ratio involving the known side, the unknown side, and either nonright angle. 2. Solve for the unknown side. ➌ Solve for c in the triangle at right sin 22 = c 2. c sin 22 = 22 b 3. c = sin Note that we could have also started with cos 68 = c, but sin 68 = b c or cos 22 = b c would not have helped because these equations have two unknowns. c

7 51 Wyner Math cademy I Spring Inverse Trigonometric Functions in Right Triangles Whereas sin, cos, and tan take an angle and find a ratio for it, the INVERSE Trigonometric Functions SIN -1, OS -1, and TN -1 take a ratio and find an angle for it. For example, sin 30 = ½ and sin -1 ½ = 30. sin -1 sin = cos -1 cos = tan -1 tan = sin -1, cos -1, and tan -1 are also called RSIN, ROS, and RTN, respectively. ➊ alculate an angle measure in a right triangle based on two known side lengths. 1. Write a sine, cosine, or tangent ratio involving the angle and the two known sides. 2. Solve for the angle by applying sin -1, cos -1, or tan -1 to each side of the equation. ➊ Solve for in the triangle at right. 1. sin = sin -1 sin = sin Note that we could have also started with cos = 8 10 or tan = To solve a triangle is to find every unknown side and angle. ngles are labeled with capital letters. Each side is labeled with the same letter as the angle opposite it, but lowercase. ➋ Solve a right triangle. 1. If both angles are unknown, solve for one of them using sin -1, cos -1, or tan -1 (see ➍). 2. Find the second angle by subtracting the first from If two sides are unknown, solve for one of them using sin, cos, or tan (see ➌). 4. Find the last side by using sin, cos, or tan, or by using the Pythagorean theorem. ➋ Solve the triangles shown below. a) b) 1. tan = 5 tan -1 tan 5 = tan -1 5 c = 67.4 = = tan 50 = e tan 50 = e sin 22.6 = cos 50 = e c sin 22.6 = 5 f cos 50 = 5 c = sin 22.6 = 13.0 f = cos f e