Students will… Write two-column proofs. Prove geometric theorems by using deductive reasoning.
Geometric Reasoning
description
Transcript of Geometric Reasoning
![Page 1: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/1.jpg)
Geometric Reasoning
Mahobe
![Page 2: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/2.jpg)
The facts to learn
![Page 3: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/3.jpg)
• Angles in a triangle add up to 180o
![Page 4: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/4.jpg)
• Angles on a straight line add up to 180o
![Page 5: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/5.jpg)
• Angles around a point add up to 360o
![Page 6: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/6.jpg)
![Page 7: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/7.jpg)
![Page 8: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/8.jpg)
![Page 9: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/9.jpg)
base ’s isos ∆ =∠
![Page 10: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/10.jpg)
![Page 11: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/11.jpg)
∠ sum quad
• Interior angles of a quadrilateral sum to 360o
![Page 12: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/12.jpg)
opp ,∠ cyclic quad
• Opposite angles of a cyclic quadrilateral sum to 180o
![Page 13: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/13.jpg)
Ext , of cyclic quad∠
• Exterior angle of a cyclic quad equals the opposite interior angle
![Page 14: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/14.jpg)
vert opp s =∠
• Vertically opposite angles are equal
![Page 15: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/15.jpg)
alt s // lines∠
• Alternate angles are equal if the lines are parallel
![Page 16: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/16.jpg)
corr s // lines∠
• Corresponding angles are equal if the lines are parallel
![Page 17: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/17.jpg)
Co-int s // lines∠
• Co-interior angles sum to 180o if the lines are parallel
![Page 18: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/18.jpg)
• Exterior angles of a polygon add up to 360o
![Page 19: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/19.jpg)
If the polygon is regular all exterior angles are the same size
• Exterior angles of a polygon add up to 360o
![Page 20: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/20.jpg)
ext regular pentagon∠
• Exterior angles of a regular pentagon is 72o
![Page 21: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/21.jpg)
Interior angles of a polygon with n sides sum to 180(n-2)
![Page 22: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/22.jpg)
Find all the other angles in this figure
![Page 23: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/23.jpg)
![Page 24: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/24.jpg)
![Page 25: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/25.jpg)
![Page 26: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/26.jpg)
![Page 27: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/27.jpg)
![Page 28: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/28.jpg)
![Page 29: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/29.jpg)
![Page 30: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/30.jpg)
Vert opp angles
![Page 31: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/31.jpg)
![Page 32: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/32.jpg)
![Page 33: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/33.jpg)
A triangle formed with 2 radii is isosceles
![Page 34: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/34.jpg)
![Page 35: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/35.jpg)
![Page 36: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/36.jpg)
![Page 37: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/37.jpg)
• The angle at the centre is twice the angle at the circumference.
![Page 38: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/38.jpg)
When writing angles use 3 letters. The middle letter is where the angle is.
![Page 39: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/39.jpg)
Find the angles u, v and wO Is the centre
![Page 40: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/40.jpg)
O Is the centre
![Page 41: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/41.jpg)
O Is the centre
![Page 42: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/42.jpg)
O Is the centre
![Page 43: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/43.jpg)
Find angle d
![Page 44: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/44.jpg)
![Page 45: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/45.jpg)
![Page 46: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/46.jpg)
Calculate the size of angle ABC
![Page 47: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/47.jpg)
![Page 48: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/48.jpg)
![Page 49: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/49.jpg)
![Page 50: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/50.jpg)
![Page 51: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/51.jpg)
![Page 52: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/52.jpg)
Calculate angle CDE
![Page 53: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/53.jpg)
![Page 54: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/54.jpg)
Calculate the size of angle HIJ
![Page 55: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/55.jpg)
![Page 56: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/56.jpg)
Calculate the size of angle c
![Page 57: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/57.jpg)
C = 96o,
• Co-int angle // lines• Corr angles // lines
• ORAngles on a lineOpp angles parallelogram
![Page 58: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/58.jpg)
Calculate the size of angle VUW
![Page 59: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/59.jpg)
![Page 60: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/60.jpg)
Shape P is a regular pentagon and shape H is a regular heptagon. Calculate the size of angle CBD
![Page 61: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/61.jpg)
Shape P is a regular pentagon and shape H is a regular heptagon. Calculate the size of angle CBD
![Page 62: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/62.jpg)
The diagram gives the angles of a playground slide. Find angle XYZ.
![Page 63: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/63.jpg)
![Page 64: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/64.jpg)
Find the size of angles NMO and OPN
![Page 65: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/65.jpg)
Find the size of angles NMO and OPN
![Page 66: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/66.jpg)
Find the size of angles PRQ and PSQ
O
![Page 67: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/67.jpg)
O
![Page 68: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/68.jpg)
![Page 69: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/69.jpg)
![Page 70: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/70.jpg)
XBC 90 Rad perp. tang
CBA 20 Angles on a line
CAB 20 Base angles isos triangle
XCB 40 Ext angle = sum opp int angles
CXB 50 Angle sum triangle
![Page 71: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/71.jpg)
![Page 72: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/72.jpg)
![Page 73: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/73.jpg)
O Is the centre of the circle.Calculate the size of angle BCD
![Page 74: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/74.jpg)
![Page 75: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/75.jpg)
O Is the centre of the circle.Calculate the size of angle WYX
![Page 76: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/76.jpg)
![Page 77: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/77.jpg)
![Page 78: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/78.jpg)
Similar triangles
![Page 79: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/79.jpg)
ABC is an isosceles triangle.O is the centre of the circle.
Calculate the size of angles BAC and ODC
![Page 80: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/80.jpg)
![Page 81: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/81.jpg)
![Page 82: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/82.jpg)
Calculate the size of angles x and y.
![Page 83: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/83.jpg)
.
![Page 84: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/84.jpg)
O is the centre of the circle.Calculate the size of angle BAC.
![Page 85: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/85.jpg)
.
![Page 86: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/86.jpg)
![Page 87: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/87.jpg)
KNML is cyclic as the opposite angles sum to 180.
![Page 88: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/88.jpg)
O is the centre of the circle.Prove angle CBY = angle CAB
![Page 89: Geometric Reasoning](https://reader035.fdocuments.in/reader035/viewer/2022062410/5681634e550346895dd3eae0/html5/thumbnails/89.jpg)
angle CBY = angle CAB