calculate the angles marked with letters
Answers
1. x=69
2. a=51, b=39
3. x=56, y=56
4. a=67
Step-by-step explanation:
1. Rectangle:
The diagonals of rectangle are congruent and bisect each other.
m\angle OAB=m\angle OBA=21^{\circ}m∠OAB=m∠OBA=21
∘
All interior angle of a rectangle are right angles. So,
m\angle OBA+x=90m∠OBA+x=90
21+x=9021+x=90
x=90-21x=90−21
x=69x=69
2. Rectangle:
Similarly,
m\angle ORS=m\angle OSR=bm∠ORS=m∠OSR=b (Definition of rectangle)
m\anlge ROS=m\angle POQ=102m\anlgeROS=m∠POQ=102 (Vertical angles)
m\angle POS+m\angle ORS+m\angle OSR=180m∠POS+m∠ORS+m∠OSR=180 (Angle sum property)
102+b+b=180102+b+b=180
2b=180-1022b=180−102
2b=782b=78
b=39b=39
a+b=90a+b=90 (Right angle)
a+39=90a+39=90
a=90-39a=90−39
a=51a=51
3. Rhombus:
Diagonals of a rhombus are perpendicular bisectors.
m\angle PLQ=90m∠PLQ=90 (Definition of rhombus)
m\angle PLQ+m\angle LPQ+m\angle LQP=180m∠PLQ+m∠LPQ+m∠LQP=180 (Angle sum property)
90+34+x=18090+34+x=180
124+x=180124+x=180
x=180-124x=180−124
x=56x=56
m\angle PQS=m\angle PSQm∠PQS=m∠PSQ (Property of isosceles triangle)
x=yx=y
y=56y=56
4. square:
Diagonals of square are angle bisector.
m\angle XCD=m\angle XCY=45m∠XCD=m∠XCY=45 (Definition of square)
m\angle YXC+m\angle CXD=180m∠YXC+m∠CXD=180 (Supplementary angles)
m\angle YXC+112=180m∠YXC+112=180
m\angle YXC=180-112m∠YXC=180−112
m\angle YXC=68m∠YXC=68
m\angle YXC+m\angle XCY+m\angle XYC=180m∠YXC+m∠XCY+m∠XYC=180 (Angle sum property)
68+45+a=18068+45+a=180
113+a=180113+a=180
a=180-113a=180−113
a=67a=67
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