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How many triangle can be drawn having its angles as 50ยฐ, 67ยฐ and 63ยฐ ? Give reasons for your answer.
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Sum of angles
Infinitely many triangles can be drawn
Let OA, OB, OC and OD are rays in the anticlockwise direction such that \(\angle \)AOB = \(\angle \)COD \( =100^o\), \(\angle \)BOC = \(82^o \) Is it true to say that AOC and BOD are lines ?
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No they are not lines
The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle.
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Let the first angle be and second angle be
Third angle
In the figure, find the value of x, if AB CD and BCED.
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(Linear pair)
(Alternate angles)
In the figure \(PQSR,\ \angle SQR=25^o,\ \angle QRT=65^o, \) find \(x \) and \( y\):
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(Alternate angles)
In
If \(x + y = s + w\), prove that AOB is a straight line.
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We know that
Since
So
In the figure, OQ bisects \(\angle \)AOB. OP is a ray opposite to ray OQ. Prove that \(\angle \)POA = \(\angle \)POB
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Let
(Linear pair)
(Linear pair)
So
In the figure, if \( x+ y=w+z\), then prove that AOB is a line.
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We know that
Similarly
In the figure, ABCD, find the value of x.
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(Cointerior angles)
(Cointerior angles)
In the figure, prove that ABEF.
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But they form alternate angle
So
(Interior angles on the same side)
From (1) and (2)
AB and CD are the bisectors of the two alternate interior angles formed by the intersection of a transversal t with parallel lines l and m (shown in the figure). Show that ABCD.
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(alt interior angles)
( AB and CD are bisectors)
But they form alternate angles
So
In the figure, DE QR and AP and BP are bisectors of \(\angle \)EAB and \(\angle \)RBA respectively. Find \( \angle\)APB.
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(Cointerior angles)
In
Prove that if two lines intersect, the vertically opposite angles are equal.
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To Prove
(Proved)
On line
On line
(Proved)
In the figure, if PQRS and \(\angle \)PXM = \(50^o \) and \(\angle \)MYS = \(120^o \), find the value of \( x\).
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(alternate angle)
In the figure, PQRS and T is any point as shown in the figure, then show that \(\angle \)PQT + \(\angle \)QTS + \( \angle \)RST = \(360^o \).
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(Cointerior angles)
(Cointerior angles)
In the figure. \(POQ \) is a line, ray \(OR \) is perpendicular in line \(PQ \). \(OS \) is another ray lying between rays \(OP \) and \(OR \). Prove that \(\angle ROS=\dfrac{1}{2}(\angle QOS\angle POS) \)
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Adding on both the sides
In the figure, \(l_1l_2 \) and \( a_1a_2\). Find the value of \(x \).
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(Cooresponding angles)
(Corresponding angles)
In the figure, find the value of x.
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In
Prove that the sum of the angles of a triangle is \(180^o \)
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(Alternate angles)
(Alternate angles)
For line
(Linear pair)
In the figure, AB and BC are two plane mirrors perpendicular to each other. Prove that the incident ray PQ is parallel to ray RS.
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So (Angle sum property)
(Angle of incidence = Angle of reflection)
Adding equation (1) and (2)
Prove that a triangle must have atleast two acute angles.
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Let us assume that only one angle have to be acute, other two angles will be right or obtuse
Let assume that both one of
Then third angle (x) which is one acute angle will be O which is not possible.
In the figure, \(\angle Q>\angle R \) and \(M \) is a point on \(QR \) such that \(PM \) is the bisector of \(\angle QPR \). If the perpendicular from \(P \) on \(QR \) meets \(QR \) at \( N\), prove that \(\angle MPN=\dfrac{1}{2}(\angle Q\angle R) \)
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In
In (exterior angle property)
But (MP is the bisector)
So
In
(Proved)
In the figure, ABCD and CDEF, Also EA \(\bot \) AB. If \(\angle \)BEF = \(40^o \), then find x, y, z
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(Corresponding angles)
(Cointerior angle)
(Cointerior angles)
In the figure, if ABCD. EF \( \bot\) CD and \(\angle \)GED = \(126^o \) find \(\angle \)AGE, \(\angle \)GEF and \(\angle \)FGE.
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(alt. interior )
In the figure, OP bisects \(\angle \)AOC, OQ bisects \(\angle \)BOC and OP \(\bot \) OQ. Show that point A,O and B are collinear
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P is a point equidistant from two lines l and m intersecting at a point A. Show that AP bisects the angle between them.
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(Given)
(Common)
In the figure, \(l  m\), show that \(\angle1+\angle2\angle3=180^o \)
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