What values of and make triangle UVW triangle GEF Both trian...

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--- ## Question 1 **Question content**: What values of \( a \) and \( b \) make \( \triangle UVW \cong \triangle GEF \)? (Both triangles are right-angled, with \( \angle W = \angle F = 90^\circ \). Sides: \( VW = 3b - 15 \), \( UV = a + 3b + 30 \) (hypotenuse of \( \triangle UVW \)), \( EF = 2b \) (leg of \( \triangle GEF \)), \( GF = 5a + b \) (hypotenuse of \( \triangle GEF \)).) **Discipline**: Mathematics (Geometry) **Grade**: High School (Geometry, typically 9th-10th grade) ### Correct answer \( a = 15 \), \( b = 15 \) ### Detailed problem-solving steps 1. **Identify Corresponding Sides**: Since \( \triangle UVW \cong \triangle GEF \), their corresponding legs and hypotenuses must be equal. - The leg \( VW \) (of \( \triangle UVW \)) corresponds to leg \( EF \) (of \( \triangle GEF \)). - The hypotenuse \( UV \) (of \( \triangle UVW \)) corresponds to hypotenuse \( GF \) (of \( \triangle GEF \)). 2. **Set Up Equations for Legs**: \( VW = EF \) Substitute the given expressions: \( 3b - 15 = 2b \) 3. **Solve for \( b \)**: Subtract \( 2b \) from both sides: \( 3b - 2b - 15 = 0 \) \( b - 15 = 0 \) \( b = 15 \) 4. **Set Up Equations for Hypotenuses**: \( UV = GF \) Substitute \( b = 15 \) and the given expressions: \( a + 3(15) + 30 = 5a + 15 \) 5. **Simplify and Solve for \( a \)**: Simplify left side: \( a + 45 + 30 = a + 75 \) Equation: \( a + 75 = 5a + 15 \) Subtract \( a \) from both sides: \( 75 = 4a + 15 \) Subtract 15 from both sides: \( 60 = 4a \) Divide by 4: \( a = 15 \) ### Knowledge points involved 1. **Congruence of Right Triangles (HL Theorem)**: - **Definition**: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent (Hypotenuse-Leg (HL) Congruence Theorem). - **Application**: Here, we use HL (or CPCTC, since congruent triangles have equal corresponding sides) to set up equations for corresponding sides. 2. **Corresponding Parts of Congruent Triangles (CPCTC)**: - **Definition**: If two triangles are congruent, their corresponding sides and angles are equal. - **Application**: We use CPCTC to equate the lengths of corresponding legs and hypotenuses. 3. **Solving Linear Equations**: - **Process**: Isolate the variable by performing inverse operations (e.g., subtraction, division) to both sides of the equation. - **Application**: Used to solve for \( b \) and \( a \) after setting up the congruence-based equations. 4. **Right Triangle Anatomy**: - **Definition**: A right triangle has one \( 90^\circ \) angle, with two legs (forming the right angle) and one hypotenuse (opposite the right angle, the longest side). - **Application**: Identified the hypotenuse and legs in each triangle to match corresponding sides for congruence. ---

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