Question

prove: $\triangle gkj \cong \triangle jhg$

1. bridges in the diagram, the vertical support beam, $\overline{bx}$, is perpendicular to the deck of the bridge. the two diagonals, $\overline{ab}$ and $\overline{cb}$, are equal in length. write a two - column proof to show that the triangles formed by the vertical support beam are congruent.

given: $\overline{bx} \perp \overline{ac}, ab = cb$
prove: $\triangle axb \cong \triangle cxb$
mixed exercises
determine whether each pair of triangles is congruent. if yes, include the theorem that applies.
4.
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Explanation:

Two - Column Proof for \(\triangle AXB\cong\triangle CXB\)

Step 1: Identify right angles

Since \(\overline{BX}\perp\overline{AC}\), by the definition of perpendicular lines, \(\angle AXB\) and \(\angle CXB\) are right angles. So, \(\angle AXB=\angle CXB = 90^{\circ}\).

Step 2: Identify common side

The side \(\overline{BX}\) is common to both \(\triangle AXB\) and \(\triangle CXB\). So, \(\overline{BX}\cong\overline{BX}\) (by the Reflexive Property of Congruence).

Step 3: Identify given congruent sides

We are given that \(AB = CB\), so \(\overline{AB}\cong\overline{CB}\) (by the definition of congruent segments).

Step 4: Apply HL Congruence Theorem

In right triangles \(\triangle AXB\) and \(\triangle CXB\), we have:

• Hypotenuse \(\overline{AB}\cong\overline{CB}\)
• Leg \(\overline{BX}\cong\overline{BX}\)

By the Hypotenuse - Leg (HL) Congruence Theorem for right triangles, if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. So, \(\triangle AXB\cong\triangle CXB\).

Statements	Reasons
2. \(\angle AXB\) and \(\angle CXB\) are right angles	Definition of perpendicular lines
3. \(\angle AXB=\angle CXB\)	All right angles are congruent
4. \(\overline{BX}\cong\overline{BX}\)	Reflexive Property of Congruence
5. \(AB = CB\)	Given
6. \(\overline{AB}\cong\overline{CB}\)	Definition of congruent segments
7. \(\triangle AXB\cong\triangle CXB\)	HL (Hypotenuse - Leg) Congruence Theorem

Answer:

The two - column proof as shown above proves that \(\triangle AXB\cong\triangle CXB\) using the HL Congruence Theorem.