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Plane Geometry (II) Question 1 Level 2 ABCDA B C D is atrapezium with both ADA D ...
Jun 1, 2024
Solution by Steps
(a)
step 1
Given that ABCDABCD is a trapezium with AD⊥ABAD \perp AB and BC⊥ABBC \perp AB, and ∠ECD=45∘\angle ECD = 45^\circ, we need to prove that △DEC\triangle DEC is a right-angled triangle
step 2
Since ∠ECD=45∘\angle ECD = 45^\circ and ∠DEC=90∘\angle DEC = 90^\circ, △DEC\triangle DEC is a right-angled triangle with ∠DCE=45∘\angle DCE = 45^\circ
(b)
step 1
To prove △EDA≡△ECB\triangle EDA \equiv \triangle ECB, we need to show that they are congruent
step 2
Note that AD∥BCAD \parallel BC and AD⊥ABAD \perp AB, BC⊥ABBC \perp AB
step 3
Since ∠EDA=∠ECB=90∘\angle EDA = \angle ECB = 90^\circ, and AD=BCAD = BC, we have △EDA≡△ECB\triangle EDA \equiv \triangle ECB by the RHS (Right angle-Hypotenuse-Side) congruence criterion
Answer
△EDA≡△ECB\triangle EDA \equiv \triangle ECB
Question 2
step 1
Given a trapezium ABCDABCD with diagonals ACAC and BDBD intersecting at point EE, we need to prove that the areas of △ADE\triangle ADE and △BCE\triangle BCE are equal
step 2
Since EE is the intersection of the diagonals, EE divides the diagonals into segments of equal length
step 3
The triangles △ADE\triangle ADE and △BCE\triangle BCE share the same height from EE to ABAB
step 4
Since the bases ADAD and BCBC are equal, the areas of △ADE\triangle ADE and △BCE\triangle BCE are equal
Answer
The areas of △ADE\triangle ADE and △BCE\triangle BCE are equal.
Question 3
step 1
Given a trapezium ABCDABCD with AD∥BCAD \parallel BC and ∠DBC=∠ACB=x∘\angle DBC = \angle ACB = x^\circ, we need to prove that ∠DBC=∠ACB\angle DBC = \angle ACB
step 2
Since AD∥BCAD \parallel BC, and ∠DBC\angle DBC and ∠ACB\angle ACB are corresponding angles, they are equal
Answer
∠DBC=∠ACB=x∘\angle DBC = \angle ACB = x^\circ
Key Concept
Trapezium properties and congruence criteria
Explanation
The key concept involves understanding the properties of trapeziums, such as parallel sides and right angles, and applying congruence criteria like RHS to prove triangle congruence.
Solution by Steps
step 1
Given that ABCDA B C D is a parallelogram with AB∥DCA B \parallel D C and AD∥BCA D \parallel B C, and the diagonal BDB D is drawn
step 2
To prove that △ABD≡△CDB\triangle A B D \equiv \triangle C D B, we need to show that the triangles are congruent
step 3
In △ABD\triangle A B D and △CDB\triangle C D B, we have: - AB=CDA B = C D (opposite sides of a parallelogram are equal) - AD=BCA D = B C (opposite sides of a parallelogram are equal) - ∠ABD=∠CDB\angle A B D = \angle C D B (alternate interior angles are equal because AB∥DCA B \parallel D C and AD∥BCA D \parallel B C)
step 4
By the Side-Angle-Side (SAS) congruence criterion, △ABD≡△CDB\triangle A B D \equiv \triangle C D B
step 5
Since △ABD≡△CDB\triangle A B D \equiv \triangle C D B, it follows that AD=CBA D = C B and AB=CDA B = C D
step 6
The property of parallelograms that we have proved is that the opposite sides of a parallelogram are equal
1 Answer
A
Key Concept
Congruence of Triangles
Explanation
To prove that two triangles are congruent, we can use the Side-Angle-Side (SAS) criterion, which states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
Solution by Steps
step 1
Given that ABCDA B C D is a quadrilateral with P,Q,R,P, Q, R, and SS being the midpoints of sides AB,BC,CD,A B, B C, C D, and DAD A respectively
step 2
To prove that △PBQ\triangle P B Q is similar to △ABC\triangle A B C, we need to show that the corresponding angles are equal and the sides are proportional
step 3
Since PP and QQ are midpoints, PQP Q is parallel to ACA C and PQ=12ACP Q = \frac{1}{2} A C
step 4
Therefore, △PBQ∼△ABC\triangle P B Q \sim \triangle A B C by the Midpoint Theorem, which states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long
step 5
Since PQ∥ACP Q \parallel A C, quadrilateral PQRSP Q R S is a parallelogram because both pairs of opposite sides are parallel
2 Answer
A
Key Concept
Midpoint Theorem
Explanation
The Midpoint Theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. This theorem helps in proving similarity and parallelism in quadrilaterals.
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