Question

below is a right triangle △mno where mo = a, on = b, and mn = c. an altitude op is constructed so mp = x and pn = y. below is the proof of the pythagorean theorem, that $a^2 + b^2 = c^2$. the proof is divided into three parts, where the title of each part indicates its main purpose. complete part a of the proof. part a: prove $a^2 = c \cdot x$ statement reason 1 ∠nom ≅ ∠opm all right angles are congruent. 2 ∠omn ≅ ∠pmo pick reason 3 △mno ~ △mop pick criterion similarity 4 $\frac{a}{x}$ = pick ratio lengths of corresponding pairs of sides of similar triangles have equal ratios. (3) 5 $a^2 = c \cdot x$ multiply both sides by a · x. (4) part b: prove $b^2 = c \cdot y$ show the steps. part c: prove $a^2 + b^2 = c^2$ show the steps.

Explanation:

Step1: Analyze ∠OMN and ∠PMO

∠OMN and ∠PMO are the same angle (common angle), so they are congruent by the Reflexive Property of Congruence.

Step2: Determine Similarity Criterion

For triangles △MNO and △MOP, we have two angles congruent: ∠NOM ≅ ∠OPM (right angles) and ∠OMN ≅ ∠PMO (common angle). So by the AA (Angle - Angle) similarity criterion, △MNO ~ △MOP.

Step3: Find Corresponding Sides Ratio

In similar triangles △MNO and △MOP, the corresponding sides are: in △MNO, side \( c \) (hypotenuse \( MN \)) corresponds to side \( a \) (hypotenuse \( MO \)) in △MOP, and side \( a \) (leg \( MO \)) corresponds to side \( x \) (leg \( MP \)) in △MOP. So the ratio of corresponding sides is \( \frac{a}{x}=\frac{c}{a} \).

Step4: Derive \( a^{2}=c\cdot x \)

From \( \frac{a}{x}=\frac{c}{a} \), cross - multiply (multiply both sides by \( a\cdot x \)): \( a\times a=c\times x \), which gives \( a^{2}=c\cdot x \).

Answer:

For step 2 (∠OMN ≅ ∠PMO), the reason is "Reflexive Property of Congruence (common angle)". For step 3 (△MNO ~ △MOP), the criterion is "AA (Angle - Angle) similarity". For step 4, the ratio is \( \frac{c}{a} \) (so \( \frac{a}{x}=\frac{c}{a} \)), and step 5 follows from cross - multiplying step 4 to get \( a^{2}=c\cdot x \). The key steps show that using angle congruence (common angle and right angles), AA similarity, and corresponding side ratios, we prove \( a^{2}=c\cdot x \).