Find the area of the triangle. You try: D C. Determine whether each of the following statements is true or false. Solve for the variables.

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1 lameda USD Geometr enchmark Stud Guide ind the area of the triangle D or all right triangles, a + b c where c is the length of the hpotenuse. 5 4 a + b c 9 + b 5 + b 5 b 5 b 44 b 9 he area of a triangle bh ( 9 )( ) ( 9) ( 6) rea of D sq. un. D. rea of D 6 sq. un. he area of the triangle is 54 square units. G.SR. Solve for the variables his is a triangle. Using theorem: leg leg 9 hpotenuse legi 9i 9 Using leg:leg:hp. ratios: leg leg 9 9 :: leg hp a b J b L 45. JKL is a right scalene triangle. rea of JKL 5 sq. un. D. Perimeter of JKL ( + ) a 0 K 0 0 un. G.SR.. Page of 9 M@WUSD (USD) 0/5/4

2 lameda USD Geometr enchmark Stud Guide 3 Solve for the variables. m n p 60 n X Z his is a triangle. Using theorem: hpotenuse i 4 m m ( short leg ) Using short leg:long leg:hpotenuse ratios: short leg : hp. : 3 : hp.: long leg. m X 30. n 6. n 6 3 ( long leg ) ( short leg ) i 3 n i 3 n 3 m 4 m n n 4 3 D. p 3 E. rea of XZ 4 3 sq. un. m n 3 We can check our answer using the Pthagorean theorem: a + b c ( ) ( ) ( ) ( )( ) G.SR.. Page of 9 M@WUSD (USD) 0/5/4

3 lameda USD Geometr enchmark Stud Guide 4 Solve for the variables in the diagram. Round answers to the nearest tenth z he sum of the interior angles of a triangle is 0 : m + m + m M. 75 L 5 z We can use right triangle trigonometr to solve for and z... sin 5 tan 5 opposite sinθ hpotenuse sin sin 64 adjacent cosθ hpotenuse z cos cos 64 z D. z sin 5 E. <. 3.9 z We can check our answer using the tangent ratio. opposite tanθ adjacent tan 64 z. tan hese values are etremel close and we should epect some error on the right side of the equation because. and 3.9 were alread rounded values. G.SR. Page 3 of 9 M@WUSD (USD) 0/5/4

4 lameda USD Geometr enchmark Stud Guide 5 Solve for the variables in the diagram. Round answers to the nearest tenth. 5 X 7 w 9 n a c m Z Using the Pthagorean heorem: 7 + n X n 5 Z + n 49 n 7 n 7 n.5 We can use right triangle trigonometr to solve for a b setting up an equation using the sine function: +. n m m tan opposite sinθ hpotenuse 7 sin a 7 sin ( sin a ) sin a D. 5 n tan + 5 c E. cos n sin m a 39.5 We can solve for w since the sum of the interior angles of a triangle is 0 : m X + m + m Z w 0 w w 50.5 w 50.5 G.SR. Page 4 of 9 M@WUSD (USD) 0/5/4

5 lameda USD Geometr enchmark Stud Guide 6 Solve for the variables in the diagram. hen find the area of the triangle. Round answers to the nearest tenth. c z 6 z his is not a right triangle, so we are going to have to use the law of sines or the law of cosines. Since we don t know an of the side lengths across from a known angle measure, we are going to have to use law of cosines using angle : c a + b ab cos ( )( ) c cos 60 c cos 60. is an acute scalene triangle. sin 30 sin 70 c 00 96cos 60 c 00 96cos 60 c 7.. sin 30 sin 70 z fter finding, we can use the law of sines to find the other variables. sin sin sin a b c D. 64 z z cos 70 + E. rea of 4 sin 70 sin sin b c sin z sin sin z sin z 0.70 ( z ) ( ) sin sin sin 0.70 z 46. z 46. sin sin a c sin sin sin z 0.0 sin z ( z ) ( ) sin sin sin z 73.7 z 73.7 rea: rea absin rea ( 6 )( ) sin 60 rea 4sin 60 rea 0. herefore the area of the triangle is 0. square units. G.SR.9, G.SR.0, G.SR. Page 5 of 9 M@WUSD (USD) 0/5/4

6 lameda USD Geometr enchmark Stud Guide 7 List as man properties as ou can about a rectangle. Since it is a parallelogram: 7 Determine which characteristic(s) are true for the quadrilaterals listed below: oth pairs of opposite sides are parallel oth pairs of opposite sides are congruent Opposite angles are congruent onsecutive angles are supplementar Diagonals bisect each other Unique to a rectangle: ll angles are congruent ( 90 ) he diagonals are congruent G.O. Parallelogram Rhombus Isosceles rapezoid. Onl one pair of parallel lines. ll sides are congruent. he diagonals bisect each other D. he diagonals are congruent ) ame as man special quadrilaterals as ou can. rapezoid, Isosceles rapezoid, Parallelogram, Rhombus, Rectangle, Square, and Kite. ame all the quadrilaterals that have the propert listed. ) oth pairs of opposite angles are congruent ) ame all the quadrilaterals that have perpendicular diagonals? Rhombus, Square, and Kite. 3) ame all the quadrilaterals whose opposite angles are congruent? ) Diagonals bisect angles Parallelogram, Rhombus, Rectangle, and Square. 3) Onl one diagonal bisects the other. G.O. Page 6 of 9 M@WUSD (USD) 0/5/4

7 lameda USD Geometr enchmark Stud Guide 9 Use the given information to answer the questions below. Given: D is an isosceles trapezoid D 9 Use the given information to answer the questions below. Given: WXZ is a square M M X a) Which segment is congruent to? D W b) Which segment is congruent to D? D Z c) Which angle is congruent to D? D D d) ame three angles that are congruent to M. a) ame three segments that are congruent to W. b) ame five angles that are congruent to Z. M M DM DM c) ame five angles that are congruent to WZ. d) ame three triangles that are congruent to WZ. G.O. Page 7 of 9 M@WUSD (USD) 0/5/4

8 lameda USD Geometr enchmark Stud Guide 0 Prove that quadrilateral D is a rectangle given the coordinates of,,, and D.,, 3 7,0 D 5,4 ( ), ( ), ( ), ( ) D 0 Given: J ( 5, 4), K ( 5, ), L(,5 ), M ( 3,), and (, ) -. KLM is a rhombus.. KLM is a square.. JKL is a parallelogram. D. JKLM is a trapezoid. - Slope of 3 D Slope of 6 4 Slope of + 4 Slope of D + oth pairs of opposite sides have the same slope making them parallel. herefore, D is a parallelogram - onsecutive sides have opposite reciprocal slopes making them perpendicular. his shows that all the interior angles are right angles making D a rectangle. - G.O. Page of 9 M@WUSD (USD) 0/5/4

9 lameda USD Geometr enchmark Stud Guide Write the equation of the circle with the given information in standard form and graph. enter: ( 3,0) Diameter: 0 units Since the diameter is 0 units, then the radius would be 5 units. 5 (, ) Given: enter: (, 4) Point on the circle: ( 0, ) ( 3,0). he radius of the circle is 9 units. he equation of the circle is ( ) ( ) Using the formula for the standard equation of a h k : circle with center (, ) ( h) + ( k ) r ( ( )) ( ) ( ) Using the distance formula: ( ) ( ) + d ( ( )) ( ) ( ) ( ) n equation of a concentric circle is ( ) ( ) D. nother point on the circle is ( 3, ). Graph the center first and then graph the points up, down, left, and right 5 units from the center and then graph the circle through those 4 points G.GPE. Page 9 of 9 M@WUSD (USD) 0/5/4

10 lameda USD Geometr enchmark Stud Guide Write the equation of the circle with the given information in standard form and graph. + + Given the equation: he equation is not in standard form, h + k r. We have to complete the square ( ) ( ) for and for to put the equation in standard form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ow the equation is in the standard form for a circle and we can see the center is ( 4, ) and r 9, which means the radius would have to be 3 units. Graph the center first and then graph the points up, down, left, and right 3 units from the center and then graph the circle through those 4 points.. he radius of the circle is units.. he center of the circle is ( 6, 3).. he equation of the circle is ( ) ( ) D. nother point on the circle is (, 3). E. Graph of the given equation: G.GPE. Page 0 of 9 M@WUSD (USD) 0/5/4

11 lameda USD Geometr enchmark Stud Guide 3 Write the equation of a parabola with a focus 0, 4 and a directri of 4. of ( ) - V : ( 0, 4) - parabola is the set of points, (, ) d d (, ), equidistant from a line (directri) and a point not on the line (focus). D : 4 3 Write the equation of a parabola with a 0, 4 and a directri of. verte of ( ). he parabola opens upward.. he equation is ( ) 4.. he equation is ( ) D. he focus is (,4). E. Graph: 4. Using the formula for a vertical parabola: Verte: ( 0,0 ) ocus: ( 0, 4) p 4 p 4 (opens down) ( ) ( ) 4 p k h Using the definition: d d ( ) ( ( )) ( ) ( 4 ) + ( + 4) ( )( ) ( ) G.GPE. Page of 9 M@WUSD (USD) 0/5/4

12 lameda USD Geometr 4 Use the diagram and given information to find the measures of the following angles and arcs. Given: G is tangent to O. G a) m m + m m m m 0 b) med me + med + md md 0 + med med med 0 c) m m ( me + m ) m ( ) m ( 90 ) m 45 d) m D m D md m D me + med m D ( ) m D ( 30 ) m D 65 e) m G m G ( m md) m G ( 0 30 ) m G ( 50 ) m G 5 0 E ( ) OR D O tangent is perpendicular to the radius at the point of tangenc. m G + m G + m G 0 m G m G m G 5 G.. 4 enchmark Stud Guide Given: KP is tangent to 00 J 30 O.. 40 H O. m KL 65. KLO is an equilateral triangle. D. mm 9 K E. m JH 30. m HM 9 M L 9 P Page of 9 M@WUSD (USD) 0/5/4

13 lameda USD Geometr enchmark Stud Guide 5 Use the diagram and given information to find the measures of the following angles and arcs. Given: O 5 Given: JK is tangent to J M O. 40 K O O z w D a) m he measure of the central angle is equal to the arc it intercepts. herefore m O m and m 90. b) md md md + m md md 30 c) m he measure of an inscribed angle is equal to half the arc it intercepts: m m m ( 90 ) m 45 d) m D m D m m D ( 0 ) m D 90 e) m D md md + md md 40 md m D md m D ( 40 ) m D 0 statements must be true... z. w D. + z 90 E. w + z 0 L G.. Page 3 of 9 M@WUSD (USD) 0/5/4

14 lameda USD Geometr enchmark Stud Guide 6 Use the diagram and given information to find the measures of the following angles and segments below. Given: O 6 Given: O, M K I 60 H J O E 53 5 O 6 L K a) m E Since the diameter bisects D, then D. herefore m E 90 b) m m m m md m m OD m ( 53 ) m 6.5 c) OE Since O is a radius, then O 5. If E, then that would make OE 3. d) D Since OED is a right triangle we can use Pthagorean theorem to find ED: 3 + ED ED 5 ED 6 ED 4 Since the diameter is perpendicular to a chord, it bisects the chord. herefore E ED 4 and D. e) E Since is a right triangle we can use Pthagorean theorem to find : D. OM. MK 0. HK 4 4 D. IJ OH E. mml 90. mmhl 0 M G.. Page 4 of 9 M@WUSD (USD) 0/5/4

15 lameda USD Geometr 7 Use the diagram to find the lengths of the segments below. S 6 a) S angent segments from a common eternal point are congruent: S S b) VM S 6 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord: VM imx ZM imw c) PW Let n equal the length of PW V 0 7 P Z M W VM i4 i7 VM i4 i7 4 4 iii7 VM i7 VM 4 If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its eternal segment equals the square of the length of the tangent segment: PW ipz ( VP) ( ) ( 0) n( n + 5) 00 ni n 4 n + 5n 00 n + 5n 00 0 ( n + 0)( n 5) 0 ( n + 0) 0 or ( n 5) 0 n 0 or n 5 X Since PW is a length, then it must be positive and therefore PW5. G.O.9 7 enchmark Stud Guide Use the diagram and given information to find the values of the variables. Given: P is tangent to O, H G M w DP is tangent to O.. w D. z 6 E D 6 4 End of Stud Guide z P Page 5 of 9 M@WUSD (USD) 0/5/4

16 lameda USD Geometr enchmark Stud Guide ou r Solutions: rea of D sq. un sin 5 tan 5 D. rea of D 6 sq. un. D. z sin 5. a b. JKL is a right scalene triangle. rea of JKL 5 sq. un. D. Perimeter of JKL ( + ) 0 0 un. 5 E. < +. n m m tan 3. m X 30. D. 5 n tan + 5 c E. cos n sin m. n 6. n 6 3 D. p 3 E. rea of XZ 4 3 sq. un. Page 6 of 9 M@WUSD (USD) 0/5/4

17 lameda USD Geometr enchmark Stud Guide 6 9 a) ame three segments that are congruent to W.. is an acute scalene triangle X,, Z.. sin 30 sin 70 sin 30 sin 70 z b) ame five angles that are congruent to Z. n five of the following angles: X, X, XW, WX, WZ, ZW, Z D. 64 z z cos 70 + E. rea of 4 sin 70 c) ame five angles that are congruent to WZ. WX, XWZ, XZ, WX, X, Z, WZ 7 Parallelogram Rhombus, Isosceles rapezoid, D d) ame three triangles that are congruent to WZ. WZ, XZ, ZX, WX, XZW, WX, ZXW ) oth pairs of opposite angles are congruent Parallelogram, rhombus, rectangle and square. 0 ) Diagonals bisect angles Rhombus and square. 3) Onl one diagonal bisects the other. Kite. KLM is a rhombus.. KLM is a square.. JKL is a parallelogram. D. JKLM is a trapezoid. Page 7 of 9 M@WUSD (USD) 0/5/4

18 lameda USD Geometr enchmark Stud Guide 3. he radius of the circle is 9 units. he parabola opens upward.. he equation of the circle is ( ) ( ) n equation of a concentric circle is ( ) ( ). he equation is ( ) 4.. he equation is ( ) D. he focus is (,4). 4. D. nother point on the circle is ( 3, ). E. Graph:. he radius of the circle is units. -. he center of the circle is ( 6, 3).. he equation of the circle is ( ) ( ) D. nother point on the circle is (, 3). E. Graph of the given equation: m KL 65. KLO is an equilateral triangle. - D. mm 9 E. m JH 30. m HM 9 - Page of 9 M@WUSD (USD) 0/5/4

19 lameda USD Geometr enchmark Stud Guide 5.. z. w D. + z 90 E. w + z 0 6. OM. MK 0. HK 4 4 D. IJ OH E. mml 90. mmhl 0 7. w D. z 6 Page 9 of 9 M@WUSD (USD) 0/5/4

Given: You try: Given: Prove: ABC ADC. Prove: ABE DCE. 1. Given 1. AB AD BC DC. 2. AC AC 2. Reflexive Prop. of. 3. ABC ADC 3. SSS Post.

Given: You try: Given: Prove: ABC ADC. Prove: ABE DCE. 1. Given 1. AB AD BC DC. 2. AC AC 2. Reflexive Prop. of. 3. ABC ADC 3. SSS Post. US Geometr enchmark Stud Guide 1 Given: 1 Given: Prove: Statements 1. 1. Given Reasons.. Refleive Prop. of 3. 3. SSS Post Prove: G.O.1 etermine whether the following statements are rue or alse. ) ) ) )