Premium Online Home Tutors

3 Tutor System

Starting just at 265/hour

(i) \(\triangle{APB}\) \(\displaystyle \cong \) \(\triangle{CQD}\)

(ii) AP = CQ

Answer :

Given: ABCD is a parallelogram and AP and CQ are perpendicular from vertices A and C on diagonal BD.

\(\because \) AB || CD and BD is a transversal, we get,

\(\angle{CDB}\) = \(\angle{DBA}\) ...(i)

Now, in \(\triangle{APB}\) and \(\triangle{CQD}\), we have,

CD = AB ...(Sides of parallelogram)

\(\angle{CQD}\) = \(\angle{APB}\) = \(90^\circ\) ...(Given)

\(\angle{CDQ}\) = \(\angle{ABP}\) ...(From Eq. (i))

\(\therefore \) \(\triangle{APB}\) \(\displaystyle \cong \) \(\triangle{CQD}\) ...(By ASA rule)

\(\therefore \) AP = CQ ...(By CPCT)

Hence, it is proved.

- The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
- If the diagonals of a parallelogram are equal, then show that it is a rectangle.
- Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
- Show that the diagonals of a square are equal and bisect each other at right angles.
- Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
- Diagonal AC of a parallelogram ABCD bisects \(\angle{A}\) (see figure). Show that (i) It bisects \(\angle{C}\) also, (ii) ABCD is a rhombus.
- ABCD is a rhombus. Show that diagonal AC bisects \(\angle{A}\) as well as \(\angle{C}\) and diagonal BD bisects \(\angle{B}\) as well as \(\angle{D}\).
- ABCD is a rectangle in which diagonal AC bisects \(\angle{A}\) as well as \(\angle{C}\). Show that: (i) ABCD is a square (ii) Diagonal BD bisects \(\angle{B}\) as well as \(\angle{D}\).
- In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see figure). Show thati) \(\triangle{APD}\) \(\displaystyle \cong \) \(\triangle{CQB}\) ii) AP = CQiii) \(\triangle{AQB}\) \(\displaystyle \cong \) \(\triangle{CPD}\) iv) AQ = CPv) APCQ is a parallelogram.
- In \(\triangle{ABC}\) and \(\triangle{DEF}\), AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F, respectively (see figure). Show that (i) Quadrilateral ABED is a parallelogram (ii) Quadrilateral BEFC is a parallelogram (iii) AD || CF and AD = CF (iv) Quadrilateral ACFD is a Parallelogram (v) AC = DF (vi) \(\triangle{ABC}\) \(\displaystyle \cong \) \(\triangle{DEF}\)
- ABCD is a trapezium in which AB || CD and AD = BC (see figure). Show that i) \(\angle{A}\) = \(\angle{B}\) ii) \(\angle{C}\) = \(\angle{D}\) iii) \(\triangle{ABC}\) \(\displaystyle \cong \) \(\triangle{BAD}\) iv) Diagonal AC = Diagonal BD [Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E].

- NCERT solutions for class 9 maths chapter 1 Number Systems
- NCERT solutions for class 9 maths chapter 2 Polynomials
- NCERT solutions for class 9 maths chapter 3 Coordinate geometry
- NCERT solutions for class 9 maths chapter 4 Linear equations in two variables
- NCERT solutions for class 9 maths chapter 5 Introduction to Euclidean Geometry
- NCERT solutions for class 9 maths chapter 6 Lines and Angles
- NCERT solutions for class 9 maths chapter 7 Triangles
- NCERT solutions for class 9 maths chapter 8 Quadrilaterals
- NCERT solutions for class 9 maths chapter 9 Areas of parallelograms and triangles
- NCERT solutions for class 9 maths chapter 10 Circles
- NCERT solutions for class 9 maths chapter 11 Constructions
- NCERT solutions for class 9 maths chapter 12 Heron's Formula
- NCERT solutions for class 9 maths chapter 13 Surface areas and volumes
- NCERT solutions for class 9 maths chapter 14 Statistics
- NCERT solutions for class 9 maths chapter 15 Probability

- NCERT solutions for class 9 science chapter 1 Matter in our Surroundings
- NCERT solutions for class 9 science chapter 2 Is Matter Around Us Pure
- NCERT solutions for class 9 science chapter 3 Atoms and Molecules
- NCERT solutions for class 9 science chapter 4 Structure of the Atom
- NCERT solutions for class 9 science chapter 5 The Fundamental Unit of Life
- NCERT solutions for class 9 science chapter 6 Tissues and Fundamental unit of life
- NCERT solutions for class 9 science chapter 7 Diversity in Living Organisms
- NCERT solutions for class 9 science chapter 8 Motion
- NCERT solutions for class 9 science chapter 9 Force and Laws of Motion
- NCERT solutions for class 9 science chapter 10 Gravitation
- NCERT solutions for class 9 science chapter 11 Work and Energy
- NCERT solutions for class 9 science chapter 12 sound
- NCERT solutions for class 9 science chapter 13 Why do We Fall Ill
- NCERT solutions for class 9 science chapter 14 Natural Resources
- NCERT solutions for class 9 science chapter 15 Improvement in Food Resources